iii southern-summer school on mathematical biology...2 predation 3 lotka-volterra 4...
TRANSCRIPT
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III Southern-Summer School on Mathematical Biology
Roberto André Kraenkel, IFT
http://www.ift.unesp.br/users/kraenkel
Lecture II
São Paulo, February 2014
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 1 / 22
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Outline
1 Interacting Species
2 Predation
3 Lotka-Volterra
4 Beyond Lotka-Volterra
5 Glory and Misery of the Lotka-Volterra Equations
6 Further beyond the Lotka-Volterra equations
7 Final comments
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 2 / 22
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Outline
1 Interacting Species
2 Predation
3 Lotka-Volterra
4 Beyond Lotka-Volterra
5 Glory and Misery of the Lotka-Volterra Equations
6 Further beyond the Lotka-Volterra equations
7 Final comments
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 2 / 22
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Outline
1 Interacting Species
2 Predation
3 Lotka-Volterra
4 Beyond Lotka-Volterra
5 Glory and Misery of the Lotka-Volterra Equations
6 Further beyond the Lotka-Volterra equations
7 Final comments
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 2 / 22
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Outline
1 Interacting Species
2 Predation
3 Lotka-Volterra
4 Beyond Lotka-Volterra
5 Glory and Misery of the Lotka-Volterra Equations
6 Further beyond the Lotka-Volterra equations
7 Final comments
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 2 / 22
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Outline
1 Interacting Species
2 Predation
3 Lotka-Volterra
4 Beyond Lotka-Volterra
5 Glory and Misery of the Lotka-Volterra Equations
6 Further beyond the Lotka-Volterra equations
7 Final comments
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 2 / 22
![Page 7: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/7.jpg)
Outline
1 Interacting Species
2 Predation
3 Lotka-Volterra
4 Beyond Lotka-Volterra
5 Glory and Misery of the Lotka-Volterra Equations
6 Further beyond the Lotka-Volterra equations
7 Final comments
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 2 / 22
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Outline
1 Interacting Species
2 Predation
3 Lotka-Volterra
4 Beyond Lotka-Volterra
5 Glory and Misery of the Lotka-Volterra Equations
6 Further beyond the Lotka-Volterra equations
7 Final comments
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 2 / 22
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Interacting Species
We saw that populations ( animals, plants, bacteria, etc) do live innetworks of trophic interactions that might be quite complex .Sometimes – as we saw – certain species can be considered effectivelynon-interacting.But in many instances, not.Let us see the simplestcases of interacting species.We begin with just two species.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 3 / 22
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Interacting Species
We saw that populations ( animals, plants, bacteria, etc)
do live innetworks of trophic interactions that might be quite complex .Sometimes – as we saw – certain species can be considered effectivelynon-interacting.But in many instances, not.Let us see the simplestcases of interacting species.We begin with just two species.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 3 / 22
![Page 11: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/11.jpg)
Interacting Species
We saw that populations ( animals, plants, bacteria, etc) do live innetworks of trophic interactions that might be quite complex .
Sometimes – as we saw – certain species can be considered effectivelynon-interacting.But in many instances, not.Let us see the simplestcases of interacting species.We begin with just two species.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 3 / 22
![Page 12: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/12.jpg)
Interacting Species
We saw that populations ( animals, plants, bacteria, etc) do live innetworks of trophic interactions that might be quite complex .Sometimes – as we saw – certain species can be considered effectivelynon-interacting.
But in many instances, not.Let us see the simplestcases of interacting species.We begin with just two species.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 3 / 22
![Page 13: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/13.jpg)
Interacting Species
We saw that populations ( animals, plants, bacteria, etc) do live innetworks of trophic interactions that might be quite complex .Sometimes – as we saw – certain species can be considered effectivelynon-interacting.But in many instances, not.
Let us see the simplestcases of interacting species.We begin with just two species.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 3 / 22
![Page 14: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/14.jpg)
Interacting Species
We saw that populations ( animals, plants, bacteria, etc) do live innetworks of trophic interactions that might be quite complex .Sometimes – as we saw – certain species can be considered effectivelynon-interacting.But in many instances, not.Let us see the simplestcases of interacting species.
We begin with just two species.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 3 / 22
![Page 15: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/15.jpg)
Interacting Species
We saw that populations ( animals, plants, bacteria, etc) do live innetworks of trophic interactions that might be quite complex .Sometimes – as we saw – certain species can be considered effectivelynon-interacting.But in many instances, not.Let us see the simplestcases of interacting species.We begin with just two species.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 3 / 22
![Page 16: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/16.jpg)
Interacting Species
We saw that populations ( animals, plants, bacteria, etc) do live innetworks of trophic interactions that might be quite complex .Sometimes – as we saw – certain species can be considered effectivelynon-interacting.But in many instances, not.Let us see the simplestcases of interacting species.We begin with just two species.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 3 / 22
![Page 17: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/17.jpg)
Types of interactions
We define three types of basic interactions between species:
Predation: the presence of a species (A) is detrimental for species(B), but thepresence of (B) is favors (A). Species (A) is the predator, and (B) is its preya.
Competition: the presence of (A) is detrimental for (B) and vice-versa.
Mutualism: the presence of (A) favors (B) and vice-versa.
aActually, we are using the word predation in a wide sense. Technically, weshould call this interaction antagonism, predation being a special case of it
Nota beneThere is also the amensalism (negative for one species, neutral for the other) and thecomensalism ( positive for one species and neutral for the other). Not to speak ofneutralism.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 4 / 22
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Types of interactions
We define three types of basic interactions between species:
Predation: the presence of a species (A) is detrimental for species(B), but thepresence of (B) is favors (A). Species (A) is the predator, and (B) is its preya.
Competition: the presence of (A) is detrimental for (B) and vice-versa.
Mutualism: the presence of (A) favors (B) and vice-versa.
aActually, we are using the word predation in a wide sense. Technically, weshould call this interaction antagonism, predation being a special case of it
Nota beneThere is also the amensalism (negative for one species, neutral for the other) and thecomensalism ( positive for one species and neutral for the other). Not to speak ofneutralism.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 4 / 22
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Types of interactions
We define three types of basic interactions between species:
Predation: the presence of a species (A) is detrimental for species(B), but thepresence of (B) is favors (A). Species (A) is the predator, and (B) is its preya.
Competition: the presence of (A) is detrimental for (B) and vice-versa.
Mutualism: the presence of (A) favors (B) and vice-versa.
aActually, we are using the word predation in a wide sense. Technically, weshould call this interaction antagonism, predation being a special case of it
Nota beneThere is also the amensalism (negative for one species, neutral for the other) and thecomensalism ( positive for one species and neutral for the other). Not to speak ofneutralism.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 4 / 22
![Page 20: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/20.jpg)
Predation
Predation is a widespread interaction between species.Ecologically, it is a direct interaction.Let us now proceed to describe a mathematical model for it.This is known as the Lotka-Volterra model.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 5 / 22
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Predation
Predation is a widespread interaction between species.
Ecologically, it is a direct interaction.Let us now proceed to describe a mathematical model for it.This is known as the Lotka-Volterra model.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 5 / 22
![Page 22: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/22.jpg)
Predation
Predation is a widespread interaction between species.Ecologically, it is a direct interaction.
Let us now proceed to describe a mathematical model for it.This is known as the Lotka-Volterra model.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 5 / 22
![Page 23: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/23.jpg)
Predation
Predation is a widespread interaction between species.Ecologically, it is a direct interaction.Let us now proceed to describe a mathematical model for it.
This is known as the Lotka-Volterra model.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 5 / 22
![Page 24: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/24.jpg)
Predation
Predation is a widespread interaction between species.Ecologically, it is a direct interaction.Let us now proceed to describe a mathematical model for it.This is known as the Lotka-Volterra model.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 5 / 22
![Page 25: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/25.jpg)
Lotka and VolterraMuoiono gl’imperi, ma i teoremi d’Euclide conservano eterna giovinezza (Volterra)
Vito Volterra (1860-1940), an Italian mathematician, pro-
posed the equation now known as the Lotka-Volterra one
to undestand a problem proposed by his futer son-in-law ,
Umberto d’Ancona, who tried to explain oscillations in the
quantity of predator fishes captured at the certain ports of
the Adriatic sea.
Alfred Lotka (1880-1949),was an USA mathematician and
chemist,born in Ukraine, who tried to transpose the prin-
ciples of physical-chemistry to biology. He published his
results in a book called “Elements of Physical Biology",
dedicated to the memory of Poynting. His results are inde-
pendent from the work of Volterra.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 6 / 22
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The Lotka-Volterra equations
LetN(t) be the number of predators,V (t) the number of preys.
In what follows, a, b, c e d are positive constants
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 7 / 22
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The Lotka-Volterra equations
O number of prey will increase when there are no predators:
dV
dt= aV
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 7 / 22
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The Lotka-Volterra equations
But the presence of predators should lower the growth rate of prey:
dV
dt= V (a − bP)
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 7 / 22
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The Lotka-Volterra equations
On the other hand the population of predators should decrease in theabsence of prey :
dV
dt= V (a − bP)
dP
dt= −dP
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 7 / 22
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The Lotka-Volterra equations
and presence of prey will increase the number of predators:
dV
dt= V (a − bP)
dP
dt= P(cV − d)
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 7 / 22
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The Lotka-Volterra equations
These two coupled equations ate known asThe Lotka-Volterra equations
dV
dt= V (a − bP)
dP
dt= P(cV − d)
Let’s study them!
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 7 / 22
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Lotka-Volterra: analysis
We have nice equations.
But we do not know their solution.These equation do not have solutions in terms of elementary functions.What can we do?Two ways
I Numerical integration. What’s that?I Qualitative analysis. What’s that?
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 8 / 22
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Lotka-Volterra: analysis
We have nice equations.But we do not know their solution.
These equation do not have solutions in terms of elementary functions.What can we do?Two ways
I Numerical integration. What’s that?I Qualitative analysis. What’s that?
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 8 / 22
![Page 34: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/34.jpg)
Lotka-Volterra: analysis
We have nice equations.But we do not know their solution.These equation do not have solutions in terms of elementary functions.
What can we do?Two ways
I Numerical integration. What’s that?I Qualitative analysis. What’s that?
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 8 / 22
![Page 35: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/35.jpg)
Lotka-Volterra: analysis
We have nice equations.But we do not know their solution.These equation do not have solutions in terms of elementary functions.What can we do?
Two waysI Numerical integration. What’s that?I Qualitative analysis. What’s that?
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 8 / 22
![Page 36: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/36.jpg)
Lotka-Volterra: analysis
We have nice equations.But we do not know their solution.These equation do not have solutions in terms of elementary functions.What can we do?Two ways
I Numerical integration. What’s that?I Qualitative analysis. What’s that?
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 8 / 22
![Page 37: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/37.jpg)
Lotka-Volterra: analysis
We have nice equations.But we do not know their solution.These equation do not have solutions in terms of elementary functions.What can we do?Two ways
I Numerical integration.
What’s that?I Qualitative analysis. What’s that?
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 8 / 22
![Page 38: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/38.jpg)
Lotka-Volterra: analysis
We have nice equations.But we do not know their solution.These equation do not have solutions in terms of elementary functions.What can we do?Two ways
I Numerical integration. What’s that?
I Qualitative analysis. What’s that?
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 8 / 22
![Page 39: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/39.jpg)
Lotka-Volterra: analysis
We have nice equations.But we do not know their solution.These equation do not have solutions in terms of elementary functions.What can we do?Two ways
I Numerical integration. What’s that?I Qualitative analysis.
What’s that?
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 8 / 22
![Page 40: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/40.jpg)
Lotka-Volterra: analysis
We have nice equations.But we do not know their solution.These equation do not have solutions in terms of elementary functions.What can we do?Two ways
I Numerical integration. What’s that?I Qualitative analysis. What’s that?
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 8 / 22
![Page 41: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/41.jpg)
Lotka-Volterra: analysis
We have nice equations.But we do not know their solution.These equation do not have solutions in terms of elementary functions.What can we do?Two ways
I Numerical integration. What’s that?I Qualitative analysis. What’s that?
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 8 / 22
![Page 42: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/42.jpg)
Lotka-Volterra: qualitative analysis
Let’s get back to the equations:
dV
dt= V (a− bP)
dP
dt= P(cV − d)
The second divided by the first:
dP
dV=
P(cV − d)
V (a− bP)
So that:dP(a− bP)
P=
dV (cV − d)
V
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 9 / 22
![Page 43: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/43.jpg)
Lotka-Volterra: qualitative analysis
Let’s get back to the equations:
dV
dt= V (a− bP)
dP
dt= P(cV − d)
The second divided by the first:
dP
dV=
P(cV − d)
V (a− bP)
So that:dP(a− bP)
P=
dV (cV − d)
V
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 9 / 22
![Page 44: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/44.jpg)
Lotka-Volterra: qualitative analysis
Let’s get back to the equations:
dV
dt= V (a− bP)
dP
dt= P(cV − d)
The second divided by the first:
dP
dV=
P(cV − d)
V (a− bP)
So that:dP(a− bP)
P=
dV (cV − d)
V
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 9 / 22
![Page 45: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/45.jpg)
Lotka-Volterra: qualitative analysis
Let’s get back to the equations:
dV
dt= V (a− bP)
dP
dt= P(cV − d)
The second divided by the first:
dP
dV=
P(cV − d)
V (a− bP)
So that:dP(a− bP)
P=
dV (cV − d)
V
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 9 / 22
![Page 46: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/46.jpg)
Lotka-Volterra: qualitative analysis
Let’s get back to the equations:
dV
dt= V (a− bP)
dP
dt= P(cV − d)
The second divided by the first:
dP
dV=
P(cV − d)
V (a− bP)
So that:dP(a− bP)
P=
dV (cV − d)
V
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 9 / 22
![Page 47: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/47.jpg)
....
dP(a− bP)
P=
dV (cV − d)
V
Integrate on both sides:a lnP − bP = cV − d lnV + H
where H is a constant.
Nn other words:cV(t)− bP(t) + a lnP(t) + d lnV(t) = H
This is a relation that has to be fulfilled by the solution of the Lotka-Volterra system ofequations.
For a given value of H we can plot on the P × V plane the geometric locus of the pointsthat obey the above relation. Let’s do it!.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 10 / 22
![Page 48: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/48.jpg)
....
dP(a− bP)
P=
dV (cV − d)
V
Integrate on both sides:
a lnP − bP = cV − d lnV + H
where H is a constant.
Nn other words:cV(t)− bP(t) + a lnP(t) + d lnV(t) = H
This is a relation that has to be fulfilled by the solution of the Lotka-Volterra system ofequations.
For a given value of H we can plot on the P × V plane the geometric locus of the pointsthat obey the above relation. Let’s do it!.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 10 / 22
![Page 49: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/49.jpg)
....
dP(a− bP)
P=
dV (cV − d)
V
Integrate on both sides:a lnP − bP = cV − d lnV + H
where H is a constant.
Nn other words:cV(t)− bP(t) + a lnP(t) + d lnV(t) = H
This is a relation that has to be fulfilled by the solution of the Lotka-Volterra system ofequations.
For a given value of H we can plot on the P × V plane the geometric locus of the pointsthat obey the above relation. Let’s do it!.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 10 / 22
![Page 50: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/50.jpg)
....
dP(a− bP)
P=
dV (cV − d)
V
Integrate on both sides:a lnP − bP = cV − d lnV + H
where H is a constant.
Nn other words:cV(t)− bP(t) + a lnP(t) + d lnV(t) = H
This is a relation that has to be fulfilled by the solution of the Lotka-Volterra system ofequations.
For a given value of H we can plot on the P × V plane the geometric locus of the pointsthat obey the above relation. Let’s do it!.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 10 / 22
![Page 51: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/51.jpg)
....
dP(a− bP)
P=
dV (cV − d)
V
Integrate on both sides:a lnP − bP = cV − d lnV + H
where H is a constant.
Nn other words:cV(t)− bP(t) + a lnP(t) + d lnV(t) = H
This is a relation that has to be fulfilled by the solution of the Lotka-Volterra system ofequations.
For a given value of H we can plot on the P × V plane the geometric locus of the pointsthat obey the above relation. Let’s do it!.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 10 / 22
![Page 52: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/52.jpg)
....
dP(a− bP)
P=
dV (cV − d)
V
Integrate on both sides:a lnP − bP = cV − d lnV + H
where H is a constant.
Nn other words:cV(t)− bP(t) + a lnP(t) + d lnV(t) = H
This is a relation that has to be fulfilled by the solution of the Lotka-Volterra system ofequations.
For a given value of H we can plot on the P × V plane the geometric locus of the pointsthat obey the above relation. Let’s do it!.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 10 / 22
![Page 53: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/53.jpg)
....
dP(a− bP)
P=
dV (cV − d)
V
Integrate on both sides:a lnP − bP = cV − d lnV + H
where H is a constant.
Nn other words:cV(t)− bP(t) + a lnP(t) + d lnV(t) = H
This is a relation that has to be fulfilled by the solution of the Lotka-Volterra system ofequations.
For a given value of H we can plot on the P × V plane the geometric locus of the pointsthat obey the above relation.
Let’s do it!.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 10 / 22
![Page 54: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/54.jpg)
....
dP(a− bP)
P=
dV (cV − d)
V
Integrate on both sides:a lnP − bP = cV − d lnV + H
where H is a constant.
Nn other words:cV(t)− bP(t) + a lnP(t) + d lnV(t) = H
This is a relation that has to be fulfilled by the solution of the Lotka-Volterra system ofequations.
For a given value of H we can plot on the P × V plane the geometric locus of the pointsthat obey the above relation. Let’s do it!.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 10 / 22
![Page 55: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/55.jpg)
....
dP(a− bP)
P=
dV (cV − d)
V
Integrate on both sides:a lnP − bP = cV − d lnV + H
where H is a constant.
Nn other words:cV(t)− bP(t) + a lnP(t) + d lnV(t) = H
This is a relation that has to be fulfilled by the solution of the Lotka-Volterra system ofequations.
For a given value of H we can plot on the P × V plane the geometric locus of the pointsthat obey the above relation. Let’s do it!.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 10 / 22
![Page 56: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/56.jpg)
Phase trajectories
dV
dt= V (a − bP)
dP
dt= P(cV − d)
The phase trajectories of the Lotka-Volterra equations, witha = b = c = d = 1. Each curve corresponds to a given value of H. Thecurves obey: cV(t)− bP(t) + a lnP(t) + d lnV(t) = H
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 11 / 22
![Page 57: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/57.jpg)
Lotka-Volterra: oscillations
We call the P × V plane, the phase space.
The curves are called trajectories or the orbits.In theis case, we have closed orbits.
What do they represent?
Take a point in the phase phase.
It represents a certain number of predators and prey.
There is a trajectory passing by this point.
As time passes by, these populations evolve according go the trajectory in phase space.
After a certain amount of time, they will come back to the initial point.
This system is periodic.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 12 / 22
![Page 58: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/58.jpg)
Lotka-Volterra: oscillations
We call the P × V plane, the phase space.
The curves are called trajectories or the orbits.In theis case, we have closed orbits.
What do they represent?
Take a point in the phase phase.
It represents a certain number of predators and prey.
There is a trajectory passing by this point.
As time passes by, these populations evolve according go the trajectory in phase space.
After a certain amount of time, they will come back to the initial point.
This system is periodic.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 12 / 22
![Page 59: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/59.jpg)
Lotka-Volterra: oscillations
We call the P × V plane, the phase space.
The curves are called trajectories or the orbits.
In theis case, we have closed orbits.
What do they represent?
Take a point in the phase phase.
It represents a certain number of predators and prey.
There is a trajectory passing by this point.
As time passes by, these populations evolve according go the trajectory in phase space.
After a certain amount of time, they will come back to the initial point.
This system is periodic.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 12 / 22
![Page 60: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/60.jpg)
Lotka-Volterra: oscillations
We call the P × V plane, the phase space.
The curves are called trajectories or the orbits.In theis case, we have closed orbits.
What do they represent?
Take a point in the phase phase.
It represents a certain number of predators and prey.
There is a trajectory passing by this point.
As time passes by, these populations evolve according go the trajectory in phase space.
After a certain amount of time, they will come back to the initial point.
This system is periodic.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 12 / 22
![Page 61: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/61.jpg)
Lotka-Volterra: oscillations
We call the P × V plane, the phase space.
The curves are called trajectories or the orbits.In theis case, we have closed orbits.
What do they represent?
Take a point in the phase phase.
It represents a certain number of predators and prey.
There is a trajectory passing by this point.
As time passes by, these populations evolve according go the trajectory in phase space.
After a certain amount of time, they will come back to the initial point.
This system is periodic.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 12 / 22
![Page 62: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/62.jpg)
Lotka-Volterra: oscillations
We call the P × V plane, the phase space.
The curves are called trajectories or the orbits.In theis case, we have closed orbits.
What do they represent?
Take a point in the phase phase.
It represents a certain number of predators and prey.
There is a trajectory passing by this point.
As time passes by, these populations evolve according go the trajectory in phase space.
After a certain amount of time, they will come back to the initial point.
This system is periodic.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 12 / 22
![Page 63: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/63.jpg)
Lotka-Volterra: oscillations
We call the P × V plane, the phase space.
The curves are called trajectories or the orbits.In theis case, we have closed orbits.
What do they represent?
Take a point in the phase phase.
It represents a certain number of predators and prey.
There is a trajectory passing by this point.
As time passes by, these populations evolve according go the trajectory in phase space.
After a certain amount of time, they will come back to the initial point.
This system is periodic.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 12 / 22
![Page 64: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/64.jpg)
Lotka-Volterra: oscillations
We call the P × V plane, the phase space.
The curves are called trajectories or the orbits.In theis case, we have closed orbits.
What do they represent?
Take a point in the phase phase.
It represents a certain number of predators and prey.
There is a trajectory passing by this point.
As time passes by, these populations evolve according go the trajectory in phase space.
After a certain amount of time, they will come back to the initial point.
This system is periodic.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 12 / 22
![Page 65: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/65.jpg)
Lotka-Volterra: oscillations
We call the P × V plane, the phase space.
The curves are called trajectories or the orbits.In theis case, we have closed orbits.
What do they represent?
Take a point in the phase phase.
It represents a certain number of predators and prey.
There is a trajectory passing by this point.
As time passes by, these populations evolve according go the trajectory in phase space.
After a certain amount of time, they will come back to the initial point.
This system is periodic.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 12 / 22
![Page 66: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/66.jpg)
Lotka-Volterra: oscillations
We call the P × V plane, the phase space.
The curves are called trajectories or the orbits.In theis case, we have closed orbits.
What do they represent?
Take a point in the phase phase.
It represents a certain number of predators and prey.
There is a trajectory passing by this point.
As time passes by, these populations evolve according go the trajectory in phase space.
After a certain amount of time, they will come back to the initial point.
This system is periodic.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 12 / 22
![Page 67: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/67.jpg)
Lotka-Volterra: oscillations
We call the P × V plane, the phase space.
The curves are called trajectories or the orbits.In theis case, we have closed orbits.
What do they represent?
Take a point in the phase phase.
It represents a certain number of predators and prey.
There is a trajectory passing by this point.
As time passes by, these populations evolve according go the trajectory in phase space.
After a certain amount of time, they will come back to the initial point.
This system is periodic.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 12 / 22
![Page 68: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/68.jpg)
Lotka-Volterra: oscillations
We call the P × V plane, the phase space.
The curves are called trajectories or the orbits.In theis case, we have closed orbits.
What do they represent?
Take a point in the phase phase.
It represents a certain number of predators and prey.
There is a trajectory passing by this point.
As time passes by, these populations evolve according go the trajectory in phase space.
After a certain amount of time, they will come back to the initial point.
This system is periodic.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 12 / 22
![Page 69: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/69.jpg)
Oscillations II
Ok, the system is periodic.Let’s take a closer look.
Take a point in the P × V plane and follow it in time:
I Let us see how the variable Vevolves (prey).
I from 1 to 3 it increases.
I from 3 to 8 it decreases.
I and from 8 to 3 it increasesagain
I and so on.
The number of prey oscillates periodically in time.and the predators so the same.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 13 / 22
![Page 70: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/70.jpg)
Oscillations II
Ok, the system is periodic.
Let’s take a closer look.
Take a point in the P × V plane and follow it in time:
I Let us see how the variable Vevolves (prey).
I from 1 to 3 it increases.
I from 3 to 8 it decreases.
I and from 8 to 3 it increasesagain
I and so on.
The number of prey oscillates periodically in time.and the predators so the same.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 13 / 22
![Page 71: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/71.jpg)
Oscillations II
Ok, the system is periodic.Let’s take a closer look.
Take a point in the P × V plane and follow it in time:
I Let us see how the variable Vevolves (prey).
I from 1 to 3 it increases.
I from 3 to 8 it decreases.
I and from 8 to 3 it increasesagain
I and so on.
The number of prey oscillates periodically in time.and the predators so the same.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 13 / 22
![Page 72: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/72.jpg)
Oscillations II
Ok, the system is periodic.Let’s take a closer look.
Take a point in the P × V plane and follow it in time:
I Let us see how the variable Vevolves (prey).
I from 1 to 3 it increases.
I from 3 to 8 it decreases.
I and from 8 to 3 it increasesagain
I and so on.
The number of prey oscillates periodically in time.and the predators so the same.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 13 / 22
![Page 73: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/73.jpg)
Oscillations II
Ok, the system is periodic.Let’s take a closer look.
Take a point in the P × V plane and follow it in time:
I Let us see how the variable Vevolves (prey).
I from 1 to 3 it increases.
I from 3 to 8 it decreases.
I and from 8 to 3 it increasesagain
I and so on.
The number of prey oscillates periodically in time.and the predators so the same.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 13 / 22
![Page 74: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/74.jpg)
Oscillations II
Ok, the system is periodic.Let’s take a closer look.
Take a point in the P × V plane and follow it in time:
I Let us see how the variable Vevolves (prey).
I from 1 to 3 it increases.
I from 3 to 8 it decreases.
I and from 8 to 3 it increasesagain
I and so on.
The number of prey oscillates periodically in time.and the predators so the same.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 13 / 22
![Page 75: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/75.jpg)
Oscillations II
Ok, the system is periodic.Let’s take a closer look.
Take a point in the P × V plane and follow it in time:
I Let us see how the variable Vevolves (prey).
I from 1 to 3 it increases.
I from 3 to 8 it decreases.
I and from 8 to 3 it increasesagain
I and so on.
The number of prey oscillates periodically in time.and the predators so the same.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 13 / 22
![Page 76: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/76.jpg)
Oscillations II
Ok, the system is periodic.Let’s take a closer look.
Take a point in the P × V plane and follow it in time:
I Let us see how the variable Vevolves (prey).
I from 1 to 3 it increases.
I from 3 to 8 it decreases.
I and from 8 to 3 it increasesagain
I and so on.
The number of prey oscillates periodically in time.and the predators so the same.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 13 / 22
![Page 77: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/77.jpg)
Oscillations II
Ok, the system is periodic.Let’s take a closer look.
Take a point in the P × V plane and follow it in time:
I Let us see how the variable Vevolves (prey).
I from 1 to 3 it increases.
I from 3 to 8 it decreases.
I and from 8 to 3 it increasesagain
I and so on.
The number of prey oscillates periodically in time.and the predators so the same.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 13 / 22
![Page 78: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/78.jpg)
Oscillations II
Ok, the system is periodic.Let’s take a closer look.
Take a point in the P × V plane and follow it in time:
I Let us see how the variable Vevolves (prey).
I from 1 to 3 it increases.
I from 3 to 8 it decreases.
I and from 8 to 3 it increasesagain
I and so on.
The number of prey oscillates periodically in time.and the predators so the same.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 13 / 22
![Page 79: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/79.jpg)
Oscillations II
Ok, the system is periodic.Let’s take a closer look.
Take a point in the P × V plane and follow it in time:
I Let us see how the variable Vevolves (prey).
I from 1 to 3 it increases.
I from 3 to 8 it decreases.
I and from 8 to 3 it increasesagain
I and so on.
The number of prey oscillates periodically in time.
and the predators so the same.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 13 / 22
![Page 80: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/80.jpg)
Oscillations II
Ok, the system is periodic.Let’s take a closer look.
Take a point in the P × V plane and follow it in time:
I Let us see how the variable Vevolves (prey).
I from 1 to 3 it increases.
I from 3 to 8 it decreases.
I and from 8 to 3 it increasesagain
I and so on.
The number of prey oscillates periodically in time.and the predators so the same.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 13 / 22
![Page 81: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/81.jpg)
Oscillations II
Ok, the system is periodic.Let’s take a closer look.
Take a point in the P × V plane and follow it in time:
I Let us see how the variable Vevolves (prey).
I from 1 to 3 it increases.
I from 3 to 8 it decreases.
I and from 8 to 3 it increasesagain
I and so on.
The number of prey oscillates periodically in time.and the predators so the same.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 13 / 22
![Page 82: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/82.jpg)
What about the solutions
Until now we saw how the solutions of the de Lotka-Volterra equations behavequalitatively.That’s a lot: we can predict that the “predator-prey” system presents periodic oscillationsof the species populations.
But, and solutions ?. The real thing!.
We can show a plot of them . Where does it come from? Numerical integration.
Here it is:
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 14 / 22
![Page 83: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/83.jpg)
What about the solutions
Until now we saw how the solutions of the de Lotka-Volterra equations behavequalitatively.
That’s a lot: we can predict that the “predator-prey” system presents periodic oscillationsof the species populations.
But, and solutions ?. The real thing!.
We can show a plot of them . Where does it come from? Numerical integration.
Here it is:
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 14 / 22
![Page 84: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/84.jpg)
What about the solutions
Until now we saw how the solutions of the de Lotka-Volterra equations behavequalitatively.That’s a lot:
we can predict that the “predator-prey” system presents periodic oscillationsof the species populations.
But, and solutions ?. The real thing!.
We can show a plot of them . Where does it come from? Numerical integration.
Here it is:
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 14 / 22
![Page 85: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/85.jpg)
What about the solutions
Until now we saw how the solutions of the de Lotka-Volterra equations behavequalitatively.That’s a lot: we can predict that the “predator-prey” system presents periodic oscillationsof the species populations.
But, and solutions ?. The real thing!.
We can show a plot of them . Where does it come from? Numerical integration.
Here it is:
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 14 / 22
![Page 86: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/86.jpg)
What about the solutions
Until now we saw how the solutions of the de Lotka-Volterra equations behavequalitatively.That’s a lot: we can predict that the “predator-prey” system presents periodic oscillationsof the species populations.
But,
and solutions ?. The real thing!.
We can show a plot of them . Where does it come from? Numerical integration.
Here it is:
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 14 / 22
![Page 87: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/87.jpg)
What about the solutions
Until now we saw how the solutions of the de Lotka-Volterra equations behavequalitatively.That’s a lot: we can predict that the “predator-prey” system presents periodic oscillationsof the species populations.
But, and solutions ?.
The real thing!.
We can show a plot of them . Where does it come from? Numerical integration.
Here it is:
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 14 / 22
![Page 88: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/88.jpg)
What about the solutions
Until now we saw how the solutions of the de Lotka-Volterra equations behavequalitatively.That’s a lot: we can predict that the “predator-prey” system presents periodic oscillationsof the species populations.
But, and solutions ?. The real thing!.
We can show a plot of them . Where does it come from? Numerical integration.
Here it is:
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 14 / 22
![Page 89: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/89.jpg)
What about the solutions
Until now we saw how the solutions of the de Lotka-Volterra equations behavequalitatively.That’s a lot: we can predict that the “predator-prey” system presents periodic oscillationsof the species populations.
But, and solutions ?. The real thing!.
We can show a plot of them .
Where does it come from? Numerical integration.
Here it is:
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 14 / 22
![Page 90: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/90.jpg)
What about the solutions
Until now we saw how the solutions of the de Lotka-Volterra equations behavequalitatively.That’s a lot: we can predict that the “predator-prey” system presents periodic oscillationsof the species populations.
But, and solutions ?. The real thing!.
We can show a plot of them . Where does it come from?
Numerical integration.
Here it is:
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 14 / 22
![Page 91: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/91.jpg)
What about the solutions
Until now we saw how the solutions of the de Lotka-Volterra equations behavequalitatively.That’s a lot: we can predict that the “predator-prey” system presents periodic oscillationsof the species populations.
But, and solutions ?. The real thing!.
We can show a plot of them . Where does it come from? Numerical integration.
Here it is:
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 14 / 22
![Page 92: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/92.jpg)
What about the solutions
Until now we saw how the solutions of the de Lotka-Volterra equations behavequalitatively.That’s a lot: we can predict that the “predator-prey” system presents periodic oscillationsof the species populations.
But, and solutions ?. The real thing!.
We can show a plot of them . Where does it come from? Numerical integration.
Here it is:
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 14 / 22
![Page 93: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/93.jpg)
What about the solutions
Until now we saw how the solutions of the de Lotka-Volterra equations behavequalitatively.That’s a lot: we can predict that the “predator-prey” system presents periodic oscillationsof the species populations.
But, and solutions ?. The real thing!.
We can show a plot of them . Where does it come from? Numerical integration.
Here it is:
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 14 / 22
![Page 94: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/94.jpg)
More about Lotka-Volterra
In words...Lotka-Volterra equations tell us that:
I Given a small number of predators and a certain number ( not small)of prey ;
I The availability of prey makes the population of predators grow;I And therefore the prey population will grow slower. After a certain
amount of time, it will begin to decrease ;I And predators attain a maximal population, and – because the lack of
enough prey – it’s population begins to decrease;I Meanwhile, prey get to a minimum and begin to recover, as the
number of predators has decreased;I and so on....
Makes sense!But, is it true?
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 15 / 22
![Page 95: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/95.jpg)
More about Lotka-Volterra
In words...Lotka-Volterra equations tell us that:
I Given a small number of predators and a certain number ( not small)of prey ;
I The availability of prey makes the population of predators grow;I And therefore the prey population will grow slower. After a certain
amount of time, it will begin to decrease ;I And predators attain a maximal population, and – because the lack of
enough prey – it’s population begins to decrease;I Meanwhile, prey get to a minimum and begin to recover, as the
number of predators has decreased;I and so on....
Makes sense!But, is it true?
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 15 / 22
![Page 96: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/96.jpg)
More about Lotka-Volterra
In words...Lotka-Volterra equations tell us that:
I Given a small number of predators and a certain number ( not small)of prey ;
I The availability of prey makes the population of predators grow;I And therefore the prey population will grow slower. After a certain
amount of time, it will begin to decrease ;I And predators attain a maximal population, and – because the lack of
enough prey – it’s population begins to decrease;I Meanwhile, prey get to a minimum and begin to recover, as the
number of predators has decreased;I and so on....
Makes sense!
But, is it true?
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 15 / 22
![Page 97: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/97.jpg)
More about Lotka-Volterra
In words...Lotka-Volterra equations tell us that:
I Given a small number of predators and a certain number ( not small)of prey ;
I The availability of prey makes the population of predators grow;I And therefore the prey population will grow slower. After a certain
amount of time, it will begin to decrease ;I And predators attain a maximal population, and – because the lack of
enough prey – it’s population begins to decrease;I Meanwhile, prey get to a minimum and begin to recover, as the
number of predators has decreased;I and so on....
Makes sense!But, is it true?
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 15 / 22
![Page 98: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/98.jpg)
The real world
Does the Lotka-Volterra equations describe real situations?
Partially.
There are some elements that are clearly not realistic:I The growth of prey in the absence of predator is exponential; it does not saturate.
F No big deal. Just put a logistic term there. We can still have oscillating solutions. Great!
I On the other hand... the growth rate of the predator is given by (cV − d).I The larger V , the higher the rate. This predator is voracious!I It would be rather natural to suppose that the conversion rate also saturates. An
effect of the predators becoming satiated or because there is handling time toconsume prey.
I We can modify the above equations to take this into account.
Cycling can still be present.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 16 / 22
![Page 99: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/99.jpg)
The real world
Does the Lotka-Volterra equations describe real situations?
Partially.
There are some elements that are clearly not realistic:I The growth of prey in the absence of predator is exponential; it does not saturate.
F No big deal. Just put a logistic term there. We can still have oscillating solutions. Great!
I On the other hand... the growth rate of the predator is given by (cV − d).I The larger V , the higher the rate. This predator is voracious!I It would be rather natural to suppose that the conversion rate also saturates. An
effect of the predators becoming satiated
or because there is handling time toconsume prey.
I We can modify the above equations to take this into account.
Cycling can still be present.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 16 / 22
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The real world
Does the Lotka-Volterra equations describe real situations?
Partially.
There are some elements that are clearly not realistic:I The growth of prey in the absence of predator is exponential; it does not saturate.
F No big deal. Just put a logistic term there. We can still have oscillating solutions. Great!
I On the other hand... the growth rate of the predator is given by (cV − d).I The larger V , the higher the rate. This predator is voracious!I It would be rather natural to suppose that the conversion rate also saturates. An
effect of the predators becoming satiated or because there is handling time toconsume prey.
I We can modify the above equations to take this into account.
Cycling can still be present.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 16 / 22
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Glory and Misery of the Lotka-Volterra Equations
GloryThe lesson of the Lotka-Volterra equation is: although being anoversimplified equation for predator-prey system it captures animportant feature: this kind of system exhibits oscillations – which areintrinsic to the dynamics.
MiserySay you are on a certain orbit in the phase space,It has certain amplitude and period.
If you perturb this orbit, the system will stay on anew orbit, with different amplitude and period.
Meaning: Bad news.Why?
As real systems are under perturbations all thetime, they would jump between trajectories andwould not be effectively periodic.
Real predator-prey oscillations would be betterdescribed by limit-cycles. What’s a limit cycle???
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 17 / 22
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Glory and Misery of the Lotka-Volterra Equations
GloryThe lesson of the Lotka-Volterra equation is: although being anoversimplified equation for predator-prey system it captures animportant feature: this kind of system exhibits oscillations – which areintrinsic to the dynamics.
MiserySay you are on a certain orbit in the phase space,It has certain amplitude and period.
If you perturb this orbit, the system will stay on anew orbit, with different amplitude and period.
Meaning:
Bad news.Why?
As real systems are under perturbations all thetime, they would jump between trajectories andwould not be effectively periodic.
Real predator-prey oscillations would be betterdescribed by limit-cycles. What’s a limit cycle???
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 17 / 22
![Page 103: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/103.jpg)
Glory and Misery of the Lotka-Volterra Equations
GloryThe lesson of the Lotka-Volterra equation is: although being anoversimplified equation for predator-prey system it captures animportant feature: this kind of system exhibits oscillations – which areintrinsic to the dynamics.
MiserySay you are on a certain orbit in the phase space,It has certain amplitude and period.
If you perturb this orbit, the system will stay on anew orbit, with different amplitude and period.
Meaning: Bad news.
Why?
As real systems are under perturbations all thetime, they would jump between trajectories andwould not be effectively periodic.
Real predator-prey oscillations would be betterdescribed by limit-cycles. What’s a limit cycle???
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 17 / 22
![Page 104: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/104.jpg)
Glory and Misery of the Lotka-Volterra Equations
GloryThe lesson of the Lotka-Volterra equation is: although being anoversimplified equation for predator-prey system it captures animportant feature: this kind of system exhibits oscillations – which areintrinsic to the dynamics.
MiserySay you are on a certain orbit in the phase space,It has certain amplitude and period.
If you perturb this orbit, the system will stay on anew orbit, with different amplitude and period.
Meaning: Bad news.Why?
As real systems are under perturbations all thetime, they would jump between trajectories andwould not be effectively periodic.
Real predator-prey oscillations would be betterdescribed by limit-cycles. What’s a limit cycle???
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 17 / 22
![Page 105: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/105.jpg)
Glory and Misery of the Lotka-Volterra Equations
GloryThe lesson of the Lotka-Volterra equation is: although being anoversimplified equation for predator-prey system it captures animportant feature: this kind of system exhibits oscillations – which areintrinsic to the dynamics.
MiserySay you are on a certain orbit in the phase space,It has certain amplitude and period.
If you perturb this orbit, the system will stay on anew orbit, with different amplitude and period.
Meaning: Bad news.Why?
As real systems are under perturbations all thetime, they would jump between trajectories andwould not be effectively periodic.
Real predator-prey oscillations would be betterdescribed by limit-cycles. What’s a limit cycle???
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 17 / 22
![Page 106: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/106.jpg)
Glory and Misery of the Lotka-Volterra Equations
GloryThe lesson of the Lotka-Volterra equation is: although being anoversimplified equation for predator-prey system it captures animportant feature: this kind of system exhibits oscillations – which areintrinsic to the dynamics.
MiserySay you are on a certain orbit in the phase space,It has certain amplitude and period.
If you perturb this orbit, the system will stay on anew orbit, with different amplitude and period.
Meaning: Bad news.Why?
As real systems are under perturbations all thetime, they would jump between trajectories andwould not be effectively periodic.
Real predator-prey oscillations would be betterdescribed by limit-cycles.
What’s a limit cycle???
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 17 / 22
![Page 107: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/107.jpg)
Glory and Misery of the Lotka-Volterra Equations
GloryThe lesson of the Lotka-Volterra equation is: although being anoversimplified equation for predator-prey system it captures animportant feature: this kind of system exhibits oscillations – which areintrinsic to the dynamics.
MiserySay you are on a certain orbit in the phase space,It has certain amplitude and period.
If you perturb this orbit, the system will stay on anew orbit, with different amplitude and period.
Meaning: Bad news.Why?
As real systems are under perturbations all thetime, they would jump between trajectories andwould not be effectively periodic.
Real predator-prey oscillations would be betterdescribed by limit-cycles. What’s a limit cycle???
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 17 / 22
![Page 108: III Southern-Summer School on Mathematical Biology...2 Predation 3 Lotka-Volterra 4 BeyondLotka-Volterra 5 GloryandMiseryoftheLotka-VolterraEquations 6 FurtherbeyondtheLotka-Volterraequations](https://reader034.vdocument.in/reader034/viewer/2022043008/5f95fa05be90845b8f705f6b/html5/thumbnails/108.jpg)
Further beyond the Lotka-Volterra equations
Obviously real interactions occur in interaction webs that can involve many speciesthrough predation, competition and mutualism.
Simple questions:I Whereupon does the prey feed?I This is not taken into account in the Lotka-Volterra equations.I If resource availability for prey is approximatively constant than a (generalized)
Lotka-Volterra dynamics is maybe a good model.I But, on the other hand, the possibility exists that the prey and its resource are
dynamically coupled... In this case we need to consider at least three species.I But beware!!! Do not try to put all species in a model.
In summary, the Lotka-Volterra equations are rather a staring point than a finalpoint for predator-prey models. .
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 18 / 22
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Further beyond the Lotka-Volterra equations
Obviously real interactions occur in interaction webs that can involve many speciesthrough predation, competition and mutualism.
Simple questions:I Whereupon does the prey feed?I This is not taken into account in the Lotka-Volterra equations.I If resource availability for prey is approximatively constant than a (generalized)
Lotka-Volterra dynamics is maybe a good model.I But, on the other hand, the possibility exists that the prey and its resource are
dynamically coupled... In this case we need to consider at least three species.I But beware!!! Do not try to put all species in a model.
In summary, the Lotka-Volterra equations are rather a staring point than a finalpoint for predator-prey models. .
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 18 / 22
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A last comment
Host-parasitoid relationsIn close relation to the predator-prey dynamics there is the relation aparasitoid and its host ,The parasitoid plays a role analogous to the one of the predator andthe host, that of the prey.Although these may be seen as different biological interactions, thedynamics is similarly described.
Note, however, that many insect species have non-overlapinggenerations.which takes us to the realm of discrete-time equations, or coupledmappings.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 19 / 22
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A last comment
Host-parasitoid relationsIn close relation to the predator-prey dynamics there is the relation aparasitoid and its host ,The parasitoid plays a role analogous to the one of the predator andthe host, that of the prey.Although these may be seen as different biological interactions, thedynamics is similarly described.Note, however, that many insect species have non-overlapinggenerations.which takes us to the realm of discrete-time equations, or coupledmappings.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 19 / 22
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What I should remember
Two-species interactions are the building blocks of larger networks ofinteractions:In a rough way, we can divide them as:
I predator-prey;I competition;I mutualism.
Predator-Prey tend to produce oscillations.Just don’t forget that not every oscillation comes from a predator-preydynamics.
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 20 / 22
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Bibliography
J.D. Murray: Mathematical Biology I (Springer, 2002)G.E. Hutchinson: An Introduction to Population Ecology ( Yale,1978).N.F. Britton: Essential Mathematical Biology ( Springer, 2003).N.J. Gotelli: A Primer of Ecology ( Sinauer, 2001).E. Renshaw: Modelling Biological Populations in Space and Time (CUP 2001).M. Kot: Elements of Mathematical Ecology (CUP 2001).P. Turchin Complex population Dynamics (Princeton U.P 2003)
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 21 / 22
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Online Resources
http://www.ictp-saifr.org/mathbio3http://ecologia.ib.usp.br/ssmb/
Thank you for your attention
Roberto A. Kraenkel (IFT-UNESP) III SSSMB São Paulo, Feb 2014 22 / 22