zdenˇek posp´ıˇsil masaryk university, faculty of …pospisil/files/sscb2011.pdf · masaryk...

Population dynamics – a source of diversity

Zdenek Pospısil

Masaryk University, Faculty of scienceDepartment of Mathematics and Statistics

7th Summer School on Computational Biology

September 15, 2011

Introduction

General theory ofpopulation dynamics

Models with discretetime

Models withcontinuous time

Population dynamics – 2 / 18

Introduction

Introduction

A bit of history

Sources, textbooks

Software





A bit of history

Introduction

A bit of history

Sources, textbooks

Software





Leonhard Euler: Introductio in analysin infinitorum (1748)

Geometric growth of population

A bit of history

Introduction

A bit of history

Sources, textbooks

Software





Thomas Robert Malthus: An Essay on the Principle ofPopulation (1798)

• The increase of population is necessarily limited by the meansof subsistence,• population does invariably increase when the means ofsubsistence increase,• the superior power of population is repressed, and the actualpopulation kept equal to the means of subsistence, by misery andvice.

A bit of history

Introduction

A bit of history

Sources, textbooks

Software





Pierre-Francois Verhulst: Recherches mathematiques sur la loied’accroissement de la population (1845)

Self-limited (logistic) growth of population

A bit of history

Introduction

A bit of history

Sources, textbooks

Software





Alfred J. Lotka: Analytical note on certain rhythmic relations inorganic systems (1920)

A bit of history

Introduction

A bit of history

Sources, textbooks

Software





Alfred J. Lotka: Analytical note on certain rhythmic relations inorganic systems (1920)Vito Volterra: Variazioni e fluttuazioni del numero d’individui inspecie animali conviventi. (1926)

A bit of history

Introduction

A bit of history

Sources, textbooks

Software





Alfred J. Lotka: Elements of physical biology (1925)Vito Volterra: Lecons sur la theorie mathematique de la lutte purla vie (1931)

A bit of history

Introduction

A bit of history

Sources, textbooks

Software





Alfred J. Lotka: Elements of physical biology (1925)Vito Volterra: Lecons sur la theorie mathematique de la lutte purla vie (1931)Georgij F. Gause: The Struggle for existence (1934)

A bit of history

Introduction

A bit of history

Sources, textbooks

Software





Alfred J. Lotka: Elements of physical biology (1925)Vito Volterra: Lecons sur la theorie mathematique de la lutte purla vie (1931)Georgij F. Gause: The Struggle for existence (1934)Andrej N. Kolmogorov: Sula teoria di Volterra della lota perl’esistenza (1936)

A bit of history

Introduction

A bit of history

Sources, textbooks

Software





Charles S. Elton: Animal ecology (1927)

food chain, ecological niche, pyramid of numbers

A bit of history

Introduction

A bit of history

Sources, textbooks

Software





Charles S. Elton: Animal ecology (1927)Alexander J. Nicholson: The balance of animal population (1933)

density (in)dependent growth

A bit of history

Introduction

A bit of history

Sources, textbooks

Software






Ch. S. Elton, A. J. Nicholson: The ten-year cycle in numbers oflynx in Canada (1942)

A bit of history

Introduction

A bit of history

Sources, textbooks

Software







Ernest T. Seton: The arctic prairies (1912)

A bit of history

Introduction

A bit of history

Sources, textbooks

Software







Crawford S. Holling: The functional response of predator to preydensity and its role in mimicry and population regulation (1965)

A bit of history

Introduction

A bit of history

Sources, textbooks

Software





Patrick H. Leslie: On the use of matrices in certain populationmathematics (1945)

structured population

A bit of history

Introduction

A bit of history

Sources, textbooks

Software





John G. Skellam: Random Dispersal in Theoretical Populations(1951)

dispersal of population in space, random walk, biological invasion

Sources, textbooks

Introduction

A bit of history

Sources, textbooks

Software





Svire�ev, �. M. { Logofet, D. O. Usto�iqivost~biologiqeskiq soobwestv. Moskva: Nauka, 1978.

Sources, textbooks

Introduction

A bit of history

Sources, textbooks

Software





■ Pielou E. C.: An Introduction to Mathematical Ecology.J.Willey&Sons, New York, NY, 1969

Svire�ev, �. M. { Logofet, D. O. Usto�iqivost~biologiqeskiq soobwestv. Moskva: Nauka, 1978.

Sources, textbooks

Introduction

A bit of history

Sources, textbooks

Software






■ Svire�ev, �. M. { Logofet, D. O. Usto�iqivost~biologiqeskiq soobwestv. Moskva: Nauka, 1978.

Sources, textbooks

Introduction

A bit of history

Sources, textbooks

Software






■ Svire�ev, �. M. { Logofet, D. O. Usto�iqivost~biologiqeskiq soobwestv. Moskva: Nauka, 1978.■ Kot M.: Elements of Mathematical Ecology. Cambridge

University Press: Cambridge, New York, Melbourne,Madrid, Cape Town, Singapore, Sao Paulo, 2001

Sources, textbooks

Introduction

A bit of history

Sources, textbooks

Software








■ Thieme H. R.: Mathematics in Population Biology.Princeton University Press: Princeton and Oxford, 2003

Sources, textbooks

Introduction

A bit of history

Sources, textbooks

Software









■ Turchin P.: Complex Population Dynamics: atheoretical/empirical synthesis. Princeton University Press:Princeton, NJ, 2003

Sources, textbooks

Introduction

A bit of history

Sources, textbooks

Software









■ Turchin P.: Complex Population Dynamics: atheoretical/empirical synthesis. Princeton University Press:Princeton, NJ, 2003

■ Tkadlec E.: Populacnı ekologie. Struktura, rust a dynamikapopulacı. Univerzita Palackeho v Olomouci, Olomouc 2008

Sources, textbooks

Introduction

A bit of history

Sources, textbooks

Software





■ Edelstein-Keshet L.: Mathematical Models in Biology.Random House: New York, NY, 1988

Sources, textbooks

Introduction

A bit of history

Sources, textbooks

Software






■ Murray J. D.: Mathematical Biology I: An Introduction.Springer-Verlag: Berlin, Heidelberg, 2001

Sources, textbooks

Introduction

A bit of history

Sources, textbooks

Software







■ Britton N. F.: Essential Mathematical Biology.Springer-Verlag: London, 2003

Sources, textbooks

Introduction

A bit of history

Sources, textbooks

Software







■ Britton N. F.: Essential Mathematical Biology.Springer-Verlag: London, 2003

■ de Vries G., Hillen T., Lewis M., Muller J., Schonfisch B.:A course in Mathematical Biology. Siam: Philadelphia,2006

Software

Introduction

A bit of history

Sources, textbooks

Software





Software

Introduction

A bit of history

Sources, textbooks

Software





Kreith K., Chakerian, D.: Iterative Algebra and DynamicModeling: a curriculum for the third millennium.Springer-Verlag: New York, 1999

Software

Introduction

A bit of history

Sources, textbooks

Software





Software

Introduction

A bit of history

Sources, textbooks

Software





Henry M., Stevens H.: A Primer of Ecology with R. Springer:Dordrecht, Heidelberg, London, New York, 2009

General theory of population dynamics

Introduction

General theory ofpopulation dynamicsPrinciples ofpopulation

1st principle

2nd principle“Fundamentalequations”




Principles of population

Introduction


1st principle






Introduction


1st principle





1. A size of population sustains in the exponential increase ordecrease until environmental conditions cause a change ofthe status.


Introduction


1st principle





1. A size of population sustains in the exponential increase ordecrease until environmental conditions cause a change ofthe status.

2. The relative change of a population size equals to theimpact of environmental conditions.

1st principle

Introduction


1st principle





A size of population sustains in the exponential increase ordecrease until environmental conditions cause a change of thestatus.

x(t) = x0qt

1st principle

Introduction


1st principle






x(t) = x0qt

x(t+ 1) = x0qt+1,

1st principle

Introduction


1st principle






x(t) = x0qt

x(t+ 1) = qx(t),

1st principle

Introduction


1st principle






x(t) = x0qt

x(t+ 1) = qx(t),x(t+ 1)− x(t)

x(t)= q − 1

1st principle

Introduction


1st principle






x(t) = x0qt

x(t+ 1) = qx(t),x(t+ 1)− x(t)

x(t)= q − 1

x′(t) = x0qt ln q,

1st principle

Introduction


1st principle






x(t) = x0qt

x(t+ 1) = qx(t),x(t+ 1)− x(t)

x(t)= q − 1

x′(t) = (ln q)x(t),

1st principle

Introduction


1st principle






x(t) = x0qt

x(t+ 1) = qx(t),x(t+ 1)− x(t)

x(t)= q − 1

x′(t) = (ln q)x(t),x′(t)

x(t)= ln q

1st principle

Introduction


1st principle





q1

-1

q − 1

ln q


x(t) = x0qt

x(t+ 1) = qx(t),x(t+ 1)− x(t)

x(t)= q − 1

x′(t) = (ln q)x(t),x′(t)

x(t)= ln q

1st principle

Introduction


1st principle





q1

-1

q − 1

ln q


x(t) = x0qt

x(t+ 1) = qx(t),x(t+ 1)− x(t)

x(t)= r

x′(t) = (ln q)x(t),x′(t)

x(t)= r

2nd principle

Introduction


1st principle





The relative change of a population size equals to the impact ofenvironmental conditions.

2nd principle

Introduction


1st principle






x′(t)

x(t)= r

2nd principle

Introduction


1st principle






x′i(t)

xi(t)= i

(

x1(t), x2(t), . . . , xn(t))

, i = 1, 2, . . . , n

2nd principle

Introduction


1st principle






x′i(t)

xi(t)= i

(

x1(t), x2(t), . . . , xn(t))

, i = 1, 2, . . . , n

x(t+ 1)− x(t)

x(t)= r

2nd principle

Introduction


1st principle






x′i(t)

xi(t)= i

(

x1(t), x2(t), . . . , xn(t))

, i = 1, 2, . . . , n

xi(t+ 1)− xi(t)

xi(t)= i

(

x1(t), x2(t), . . . , xn(t))

,

i = 1, 2, . . . , n

2nd principle

Introduction


1st principle






x′i(t)

xi(t)= i

(

x1(t), x2(t), . . . , xn(t))

, i = 1, 2, . . . , n

xi(t+ 1)− xi(t)

xi(t)= i

(

x1(t), x2(t), . . . , xn(t))

,

i = 1, 2, . . . , n

x(t+ 1) = qx(t)

2nd principle

Introduction


1st principle






x′i(t)

xi(t)= i

(

x1(t), x2(t), . . . , xn(t))

, i = 1, 2, . . . , n

xi(t+ 1)− xi(t)

xi(t)= i

(

x1(t), x2(t), . . . , xn(t))

,

i = 1, 2, . . . , n

x(t+ 1) = erx(t)

2nd principle

Introduction


1st principle






x′i(t)

xi(t)= i

(

x1(t), x2(t), . . . , xn(t))

, i = 1, 2, . . . , n

xi(t+ 1)− xi(t)

xi(t)= i

(

x1(t), x2(t), . . . , xn(t))

,

i = 1, 2, . . . , n

xi(t+ 1) = ei(

x1(t),x2(t),...,xn(t))

x(t), i = 1, 2, . . . , n

“Fundamental equations”

Introduction


1st principle





Populations with overlapping generations

x′i = xii(x1, x2, . . . , xn), i = 1, 2, . . . , n

Populations with non-overlapping generations

xi(t+ 1) = ei(

x1(t),x2(t),...,xn(t))

xi(t), i = 1, 2, . . . , n


Introduction


1st principle






x′i = xii(x1, x2, . . . , xn), i = 1, 2, . . . , n


xi(t+ 1) = ei(

x1(t),x2(t),...,xn(t))

xi(t), i = 1, 2, . . . , n

i linear functions – Lotka-Volterra systemsi general functions – Kolmogorov systems


Introduction


1st principle






x′i = xi(

ri − fi(x1, x2, . . . , xn))

, i = 1, 2, . . . , n


xi(t+ 1) = eri−fi

(

x1(t),x2(t),...,xn(t))

xi(t), i = 1, 2, . . . , n

fi linear functions – Lotka-Volterra systemsfi general functions – Kolmogorov systems

f(0, 0, . . . , 0) = 0


Introduction


1st principle






x′i = xi(

ri − fi(x1, x2, . . . , xn))

, i = 1, 2, . . . , n


xi(t+ 1) = eri−fi

(

x1(t),x2(t),...,xn(t))

xi(t), i = 1, 2, . . . , n

Lotka-Volterra systems

fi(x1, x2, . . . , xn) =n∑

j=1

aijxj = Ax


Introduction


1st principle






x′i = xi(

ri − fi(x1, x2, . . . , xn))

, i = 1, 2, . . . , n


xi(t+ 1) = eri−fi

(

x1(t),x2(t),...,xn(t))

xi(t), i = 1, 2, . . . , n

Lotka-Volterra systems

fi(x1, x2, . . . , xn) =n∑

j=1

aijxj = Ax

A = (aij) – community matrix

Models with discrete time

Introduction



One populationTwo populations –Lotka-Volterrasystem



One population

Introduction






x(t+ 1) = erx(t), x(0) = x0 ⇒ x(t) = x0ert

One population

Introduction






x(t+ 1) = erx(t), x(0) = x0 ⇒ x(t) = x0ert

x(t+ 1) = ep(

x(t))

x(t), x(0) = x0

p decreasing – intra-specific competitionp increasing – Allee effect

One population

Introduction






x(t+ 1) = erx(t), x(0) = x0 ⇒ x(t) = x0ert

x(t+ 1) = ep(

x(t))

x(t), x(0) = x0


p(x) = r(

1−x

K

)

– logistic equation

x > K ⇒ ep(x) > 1, x < K ⇒ ep(x) < 1

K – carrying cappacity

One population

Introduction






x(t+ 1) = erx(t), x(0) = x0 ⇒ x(t) = x0ert

x(t+ 1) = ep(

x(t))

x(t), x(0) = x0


p(x) = r(

1−x

K

)


x > K ⇒ ep(x) > 1, x < K ⇒ ep(x) < 1

K – carrying cappacity

ep(x) =q

(1 + (q − 1)x/K)b– basic equation

Two populations – Lotka-Volterra system

Introduction






x(t+ 1) = x(t)er1−a11x(t)−a12y(t)

y(t+ 1) = y(t)er2−a21x(t)−a22y(t)


Introduction






x(t+ 1) = x(t)er1−a11x(t)−a12y(t)

y(t+ 1) = y(t)er2−a21x(t)−a22y(t)

aii > 0 – intra-specific competitionaii < 0 – Allee effectaij > 0 – inter-specific competitionaij < 0 – mutualism

aijaji < 0 – predation

Models with continuous time

Introduction




One populationTwo populations –Lotka-VolterrasystemGause-typepredator-prey system


One population

Introduction






x′ = rx, x(0) = x0 ⇒ x(t) = x0ert

One population

Introduction






x′ = rx, x(0) = x0 ⇒ x(t) = x0ert

x′ = xp(x), x(0) = x0


One population

Introduction






x′ = rx, x(0) = x0 ⇒ x(t) = x0ert

x′ = xp(x), x(0) = x0


p(x) = r(

1−x

K

)


One population

Introduction






x′ = rx, x(0) = x0 ⇒ x(t) = x0ert

x′ = xp(x), x(0) = x0


p(x) = r(

1−x

K

)


p(x) = rxa−1

(

1−( x

K

)b)

– generalized logistic equation


Introduction






x′ = x(r1 − a11x− a12y)

y′ = y(r2 − a21x− a22y)


Introduction






x′ = x(r1 − a11x− a12y)

y′ = y(r2 − a21x− a22y)

aii > 0 – intra-specific competitionaii < 0 – Allee effectaij > 0 – inter-specific competitionaij < 0 – mutualism

aijaji < 0 – predation

Gause-type predator-prey system

Introduction







Introduction






x′ = xp(x),

y′ = −dy

x – size of prey populationy – size of predator population

p(x) – prey growth rated – predator death rate


Introduction






Holling I

Holling IIIHolling IIϕ ϕ

ϕϕ

1 1

11

2a b b + 2a

xx

x x

x′ = xp(x)− Sϕ(x)y,

y′ = −dy



ϕ(x) – trophic function; increasing, ϕ(0) = 0, limx→∞

ϕ(x) = 1

S – predator level of satiety


Introduction






x′ = xp(x)− Sϕ(x)y,

y′ = −dy + κSϕ(x)y



ϕ(x) – trophic function; increasing, ϕ(0) = 0, limx→∞

ϕ(x) = 1

S – predator level of satietyκ – efficiency of conversion: prey into predator growth rate


Introduction






x′ = x

(

p(x)− Sϕ(x)

xy

)

,

y′ = y(

− d+ κSϕ(x))


Introduction






x′ = x

(

p(x)− Sϕ(x)

xy

)

,

y′ = y(

− d+ κSϕ(x))

ϕ(x) =

{

x/(2a), x < 2a

1, x ≥ 2a


Introduction






x′ = x

(

p(x)− Sϕ(x)

xy

)

,

y′ = y(

− d+ κSϕ(x))

ϕ(x) =

{

x/(2a), x < 2a

1, x ≥ 2a

ϕ(x) =xk

xk + ak

ϕ(x) = 1− 2−(x/a)k


Introduction






x′ = x

(

p(x)− Sϕ(x)

xy

)

,

y′ = y(

− d+ κSϕ(x))

ϕ(x) =

{

x/(2a), x < 2a

1, x ≥ 2a– Holling I

ϕ(x) =xk

xk + ak– Michaelis-Menten

ϕ(x) = 1− 2−(x/a)k – Ivlev

a – level of “half saturation”

zdenˇek posp´ıˇsil masaryk university, faculty of …pospisil/files/sscb2011.pdf · masaryk...

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