population dynamics, evolutionary games, and biodiversity · 2013-11-15 · population dynamics,...
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Population dynamics, evolutionary games, and biodiversity
Vlastimil Krivan Biology Center
Ceske Budejovice Czech Republic
[email protected] www.entu.cas.cz/krivan
Taken from: Mora C, Tittensor DP, Adl S, Simpson AGB, Worm B (2011) How Many Species Are There on Earth and in the Ocean? PLoS Biol 9(8): e1001127.
Biodiversity
2
• “Higher diversity, then, means longer food chains and more cases of symbiosis(mutualism, parasitism, commensalism, and so forth), and greater possibilitiesfor negative feedback control, which reduces oscillations and hence increasesstability (Odum, 1971)”
• "Complete competitors cannot coexist" (Gause 1934)
“No stable equilibrium is possible if N species are limited by less than N fac-
tors” (Levin, 1970)
3
Connectance in %
Prob
ability of stability
4
Connectance
Prob
abili
ty o
f sta
bilit
y
Connectance
Prop
or6o
n of su
rviving species
Taken from Kondoh (2003)
5
• Environmental fluctuations (relative nonlinearity of competition, Hutchin-son (1961), Armstrong and McGehee (1976), Chesson (2000),...)
• Spatial heterogeneity (metapopulation dynamics, colonization-extinctiondynamics, Hu↵aker (1958), Hanski (1999),...)
...
• Neutral theory of biodiversity (all species are ecologically identical, i.e.,they have the same fitness, and species turnover is driven by dispersal,speciation, and extinction, Hubbel 1979),
Some hypothesis for biodiversity
6
Coexistence mechanisms
• Equalizing mechanisms: tend to decrease average fitness differences between
species. These mechanisms are expressed through evolutionary/behavioral dy-
namics that change individual traits
7
8
• Population dynamics and trait dynamics operate on similar time scales
(e.g., P. Abrams, T. Vincent and J. Brown).
• Trait dynamics are very fast when compared to population dynamics (Pop-
ulation game dynamics, e.g., Cressman and VK). Assumes that traits
equilibriate at the current population abundance. The trait values are
assumed to be evolutionary optimized.
Population game dynamics
9
Herbert Spencer Charles Darwin
“Survival of the fittest” “Evolution by natural selection”
The fittest strategies do survive
10
Gi(ui;u, x) = fitness of an individual of the i � th population with strategy ui
in the resident population with strategy u:
Gi(ui;u, x) = fi(x, u)
Frequency and density dependent fitness
11
Dynamics and constraints: Viability theory
(Aubin and Cellina, 1984)
12
dx
dt
= f(x, u)
u 2 S(x)
() dx
dt
2 F (t, x) := {f(x, u) | u 2 S(x)}
ui = predator preference for prey i
(0 ui, u1 + u2 = 1)
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0 10 20 30 40 50
0.0
0.5
1.0
0 10 20 30 40 500
2
4
6
8
10
Population dynamics Patch payoffs
Patch 1
Patch 2
14
Barnacles Mussels
15
Consumer fitness is density dependent:
The optimal strategy set:
S(R1, R2) = {(p1, p2) 2 U | p1 � 0, p2 � 0, p1+p2 = 1, G(p1, p2) = max
(u1,u2)2UG(u1, u2)}
16
u1(R1, R2) 2 S(R1, R2) =
8><
>:
{1} for e1R1 > e2R2
{0} for e1R1 < e2R2
[0, 1] for e1R1 = e2R2
G(u1, u2, R1, R2) =1
C
dC
dt= e1u1R1 + e2u2R2 �m ! max
u1+u2=1
R 1
R 2
C
Optimal prey switching equalizes predator fitness in thetwo patches
0 10 20 30 40 500
1
2
3
4
5
6
0 10 20 30 40 50
-0.1
0.0
0.1
0.2
0.3
0.4
Population dynamics Predator fitness in the two patches
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§ Adaptive prey switching relaxes apparent competition between resources and
leads to their persistence
§ This is the consequence of equalizing predator fitness in the two patches.
§ There is no stabilizing mechanism in population dynamics so that the system
does not settle to an equilibrium
18
19
Patch payo↵ fi(uix) (measured as the per capita population growth rate in
patch i) depends on current population distribution
Frequency dependent fitness: The Habitat selectiongame (Cressman and VK, 2006)
20
21
22
Payo↵ in patch i: fi
= ri
(1� xiKi
)
The overall population size: x = x1 + · · ·+ xn
The IFD for the logistic population growth
(VK and Sirot, 2002; Cressman and VK 2010)
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0 5 10 15 20 25 300.6
0.7
0.8
0.9
1.0
Preferen
ce
fo
rp
atch
1(p 1
)
24 0 2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0 2 4 6 8 10
0.0
0.5
1.0
1.5
Logistic population dynamics
(Krivan and Sirot, 2002)
Population dynamics
K1 = 20, K2 = 10
Fixed preference Adaptive preference
Patch payoffs
M = m1 +m2, N = n1 + n2, pi =miM , qi =
niN
Species 1 payo↵ in habitat i : Vi(p, q;M,N) = ri
✓1� piM
Ki� ↵iqiN
Ki
◆i = 1, 2
Species 2 payo↵ in habitat j: Wj(p, q;M,N) = sj
✓1� qjN
Lj� �jpjM
Lj
◆j = 1, 2.
25
The Ideal Free Disribution for two competing populations
26
Proposition 1 (Cressman et al. 2004) Let us assume that the interior Nash equilib-
rium for the distribution of two competing species at population densities M and Nexists. If
r1s1K2L2(1�↵1�1)+r1s2K2L1(1�↵1�2)+r2s1K1L2(1�↵2�1)+r2s2K1L1(1�↵2�2) > 0
then this distribution is a 2-species ESS (2-species IFD).
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Population-distributional stability
0 10 20 30 40 50
5
10
15
20
25
30
0 10 20 30 40 500.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30 40 500.0
0.2
0.4
0.6
0.8
1.0
0 10 20 30 40 500.0
0.2
0.4
0.6
0.8
Population dynamics
28
• Conditions for population stabilization without dispersal:
1� ↵1�1 > 0, 1� ↵2�2 > 0
• Conditions for distributional stability holds (fitness equalization), i.e.,
r1s1K2L2(1�↵1�1)+r1s2K2L1(1�↵1�2)+r2s1K1L2(1�↵2�1)+r2s2K1L1(1�↵2�2) > 0
Population distribution is not an ESS
29
Population-distributional instability
Population dynamics
0 50 100 150 200
5
10
15
0 50 100 150 2000.0
0.2
0.4
0.6
0.8
1.0
0 50 100 150 200
-1.5
-1.0
-0.5
0.0
0.5
0 50 100 150 200
-1.5
-1.0
-0.5
0.0
0.5
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Population-distributional instability
Conclusion: Fitness maximization can destabilize otherwise stable population equilibrium and may not lead to fitness equalization
31
Specialized predators Generalist predators
32
Specialists Non-‐flexible generalists Flexible generalists
33
With fitness equaliza6on
Without fitness equaliza6on
34
Still more complex food webs (Berec et al, 2010)
Conclusions
35
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