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Evolutionary Games and Population Dynamics/Well-mixed populations: Difference between revisions
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Evolutionary Games and Population Dynamics/Well-mixed populations (view source)
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The negative feedback between population density and interaction group size hinges on the fact that the group size can become smaller than \(r\). For pairwise prisoner's dilemma interactions this is not the case: because \(S\) cannot vary (and is always equal to \(N=2\)), either \(r < S\) always holds (in which case the population goes extinct) or \(r>S\) always holds (in which case defectors disappear but cooperators persist). The dynamic feedback cannot operate in either case. | The negative feedback between population density and interaction group size hinges on the fact that the group size can become smaller than \(r\). For pairwise prisoner's dilemma interactions this is not the case: because \(S\) cannot vary (and is always equal to \(N=2\)), either \(r < S\) always holds (in which case the population goes extinct) or \(r>S\) always holds (in which case defectors disappear but cooperators persist). The dynamic feedback cannot operate in either case. | ||
The following figures and simulations illustrate the rich dynamics of this system. For increasing \(r\) the system undergoes a series of bifurcations. A super-critical or sub-critical [ | The following figures and simulations illustrate the rich dynamics of this system. For increasing \(r\) the system undergoes a series of bifurcations. A super-critical or sub-critical [https://en.wikipedia.org/wiki/Hopf_bifurcation Hopf bifurcation] gives rise to stable and unstable limit cycles, respectively, and a [http://www.scholarpedia.org/article/Bautin_bifurcation Bautin bifurcation] may even result in a pair of stable and unstable limit cycles that collide and disappear in a saddle-node bifurcation of periodic orbits. | ||
The phase space is spanned by the population density \(x + y\) (or \(1 - z\)) and the relative fraction of cooperators \(f = x / (x + y)\). The left boundary (\(z = 1\)) is attracting and consists of a line of stable fixed points (filled circles), which represent states where the population cannot maintain itself and disappears. Conversely, the right boundary, which denotes the maximal population density (\(z = 0\)), is repelling. In absence of cooperators (bottom boundary, \(f = 0\)), population densities decrease and eventually vanish. Finally, in absence of defectors (top boundary, \(f = 1\)), there are two saddle points (open circles) except for the last scenario where one is a stable node (filled circle). In addition, there may be an interior fixed point \(Q\) present. | The phase space is spanned by the population density \(x + y\) (or \(1 - z\)) and the relative fraction of cooperators \(f = x / (x + y)\). The left boundary (\(z = 1\)) is attracting and consists of a line of stable fixed points (filled circles), which represent states where the population cannot maintain itself and disappears. Conversely, the right boundary, which denotes the maximal population density (\(z = 0\)), is repelling. In absence of cooperators (bottom boundary, \(f = 0\)), population densities decrease and eventually vanish. Finally, in absence of defectors (top boundary, \(f = 1\)), there are two saddle points (open circles) except for the last scenario where one is a stable node (filled circle). In addition, there may be an interior fixed point \(Q\) present. | ||
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== Evolutionary scenarios == | == Evolutionary scenarios == | ||
All of the following examples and suggestions are meant as inspirations for further experimenting with the ''EvoLudo'' simulator. Each of following examples starts a lab that demonstrates the particular dynamical scenario. By modifying the [[Parameters|parameters]] the dynamics can be further explored. | All of the following examples and suggestions are meant as inspirations for further experimenting with the ''EvoLudo'' simulator. Each of following examples starts a lab that demonstrates the particular dynamical scenario. By modifying the [[Parameters|parameters]] the dynamics can be further explored. | ||
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For slightly higher \(r\) the interior fixed point \(Q\) is still an unstable focus but now surrounded by a stable limit cycle - the hallmark of a super critical Hopf bifurcation. Cooperators and defectors co-exist in never ending periodic oscillations. | For slightly higher \(r\) the interior fixed point \(Q\) is still an unstable focus but now surrounded by a stable limit cycle - the hallmark of a super critical Hopf bifurcation. Cooperators and defectors co-exist in never ending periodic oscillations. | ||
Hint: | Hint: use forward and backward integration to explore the stable limit cycle and unstable fixed points. | ||
{{-}} | {{-}} | ||
</div> | </div> | ||
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[[Image:Well-mixed Ecological PGG - stable focus.png |left|200px|link=EvoLudoLab: Ecological Public Goods Game - Stable focus]] | [[Image:Well-mixed Ecological PGG - stable focus.png |left|200px|link=EvoLudoLab: Ecological Public Goods Game - Stable focus]] | ||
==== [[EvoLudoLab: Ecological Public Goods Game - Stable focus|(d) Stable focus]]==== | ==== [[EvoLudoLab: Ecological Public Goods Game - Stable focus|(d) Stable focus]]==== | ||
Increasing \(r\) further leads to a Hopf bifurcation, the interior fixed point \(Q\) becomes a stable focus and the limit cycle disappears. Depending on the initial conditions, cooperators and defectors co-exist at some fixed densities. If exploitation by defectors is severe or population densities are too low, the population is unable to recover and goes extinct. | Increasing \(r\) further leads to a Hopf bifurcation, the interior fixed point \(Q\) becomes a stable focus and the limit cycle disappears. Depending on the initial conditions, cooperators and defectors co-exist at some fixed densities. If exploitation by defectors is severe or population densities are too low, the population is unable to recover and goes extinct. | ||
{{-}} | {{-}} | ||
</div> | </div> | ||
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[[Image:Well-mixed Ecological PGG - stable node.png |left|200px|link=EvoLudoLab: Ecological Public Goods Game - Stable node]] | [[Image:Well-mixed Ecological PGG - stable node.png |left|200px|link=EvoLudoLab: Ecological Public Goods Game - Stable node]] | ||
==== [[EvoLudoLab: Ecological Public Goods Game - Stable node|(e) Stable node]]==== | ==== [[EvoLudoLab: Ecological Public Goods Game - Stable node|(e) Stable node]]==== | ||
Another increase in \(r\) turns the interior fixed point \(Q\) into a stable node. As before, cooperators and defectors co-exist at some fixed densities only, they no longer approach the equilibrium in an oscillatory manner. Severe exploitation and low population densities again result in extinction. | Another increase in \(r\) turns the interior fixed point \(Q\) into a stable node. As before, cooperators and defectors co-exist at some fixed densities only, they no longer approach the equilibrium in an oscillatory manner. Severe exploitation and low population densities again result in extinction. | ||
{{-}} | {{-}} | ||
</div> | </div> | ||
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[[Image:Well-mixed Ecological PGG - no Q, cooperation.png |left|200px|link=EvoLudoLab: Ecological Public Goods Game - No Q, cooperation]] | [[Image:Well-mixed Ecological PGG - no Q, cooperation.png |left|200px|link=EvoLudoLab: Ecological Public Goods Game - No Q, cooperation]] | ||
==== [[EvoLudoLab: Ecological Public Goods Game - No Q, cooperation|(f) No ''Q'', cooperation]]==== | ==== [[EvoLudoLab: Ecological Public Goods Game - No Q, cooperation|(f) No ''Q'', cooperation]]==== | ||
For high \(r\), the interior fixed point \(Q\) disappears and the high density saddle node along \(f=1\), i.e. in absence of defectors, becomes a stable equilibrium. Cooperators and defectors can no longer co-exist but now its only the defectors that disappear, at least for favorable initial conditions. As always, severe exploitation and low population densities result in extinction. | For high \(r\), the interior fixed point \(Q\) disappears and the high density saddle node along \(f=1\), i.e. in absence of defectors, becomes a stable equilibrium. Cooperators and defectors can no longer co-exist but now its only the defectors that disappear, at least for favorable initial conditions. As always, severe exploitation and low population densities result in extinction. | ||
{{-}} | {{-}} | ||
</div> | </div> | ||
{{-}} | {{-}} | ||
== Complex bifurcations == | == Complex bifurcations == | ||
For larger group sizes \(N\) fascinating and much more complex Hopf bifurcations and dynamical scenarios are possible, which includes multiple, stable and unstable limit cycles. However, also note that r values for which these fascinating bifurcations occur is restricted to a tiny interval. Thus, despite their appeal from a dynamical systems' perspective, the limit cycles might be of only limited relevance for biological applications. | For larger group sizes \(N\) fascinating and much more complex Hopf bifurcations and dynamical scenarios are possible, which includes multiple, stable and unstable limit cycles. However, also note that r values for which these fascinating bifurcations occur is restricted to a tiny interval. Thus, despite their appeal from a dynamical systems' perspective, the limit cycles might be of only limited relevance for biological applications. | ||
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== Population Dynamics == | == Population Dynamics == | ||
In order to combine game dynamics and population dynamics in a replicator equation we assume that \(u\) denotes the density of cooperators, \(v\) the density of defectors and \(w=1-u-v\) the abundance of | In order to combine game dynamics and population dynamics in a replicator equation we assume that \(u\) denotes the density of cooperators, \(v\) the density of defectors and \(w=1-u-v\) represents a measure for reproductive opportunities such as the abundance of food or availability of space. Thus, \(u+v\) denotes a normalized population density such that for \(u+v=0\) (or \(w=1\)) the population has gone extinct. The dynamics of \(u, v\) and \(z\) is determined by the average payoffs (or fitness) of cooperators \(f_C\) and defectors \(f_D\) arising from game theoretical interactions. Cooperators and defectors are assumed to die at a constant rate \(d\) and give birth according to a constant baseline birth rate \(b\) augmented by their performance \(f_C\) and \(f_D\). In addition, birth events are conditional on the availability of empty space and hence are proportional to \(w\). This leads to the following population dynamic model: | ||
\begin{align*} | \begin{align*} | ||
\dot u =& u (w (b+f_C)-d)\\ | \qquad \dot u =&\ u (w (b+f_C)-d)\\ | ||
\dot v =& v (w (b+f_D)-d)\\ | \dot v =&\ v (w (b+f_D)-d)\\ | ||
\dot w =& -\dot u -\dot v | \dot w =& -\dot u -\dot v | ||
\end{align*} | \end{align*} | ||
