Evolutionary Games and Population Dynamics/Well-mixed populations - Revision history

Hauert at 15:52, 13 October 2023

2023-10-13T15:52:46Z

← Older revision		Revision as of 08:52, 13 October 2023
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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 [[Hopf bifurcation]] gives rise to stable and unstable limit cycles, respectively, and a [[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 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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	\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*}

Hauert at 17:23, 16 March 2016

2016-03-16T17:23:19Z

← Older revision		Revision as of 10:23, 16 March 2016
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	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.
			{{-}}

	~~{{-}}~~
	== 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: ~~often, the~~ forward integration ~~will not stop and keep tracking~~ the stable limit cycle~~. Just click on the phase plane to stop forward integration~~ and ~~start the backward integration. Another click stops backward integration, too~~.		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.

Hauert at 19:14, 31 March 2012

2012-03-31T19:14:32Z

← Older revision		Revision as of 12:14, 31 March 2012
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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 ~~empty~~ 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:		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*}

Hauert at 19:12, 31 March 2012

2012-03-31T19:12:23Z

← Older revision		Revision as of 12:12, 31 March 2012
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	\begin{align*}		\begin{align*}
	\dot u =& x (w (b+f_C)-d)\\		\dot u =& u (w (b+f_C)-d)\\
	\dot v =& y (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*}

Hauert at 19:11, 31 March 2012

2012-03-31T19:11:42Z

← Older revision		Revision as of 12:11, 31 March 2012
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	{{InCharge\|author1=Christoph Hauert}}		{{InCharge\|author1=Christoph Hauert}}
	__NOTOC__		__NOTOC__

	[[Image:Well-mixed Ecological PGG - Bifurcations.png\|thumb\|300px\|Bifurcations and dynamical regimes (separated by dashed lines) in heterogeneous populations of cooperators and defectors interacting in ecological public goods games with \(N=8\), \(b=1\), \(d=1.2\) and the multiplication factor \(r\) as the bifurcation parameter. The position of the fixed point \(Q\) changes with increasing \(r\) while the system undergoes a series of different bifurcations: \(Q\) enters on the top left for low \(r\) and leaves at the top right for high \(r\) (transcritical bifurcations). The different dynamical scenarios (a)-(f) apply depending on the location of \(Q\). In (a) and (f) \(Q\) is absent, in (b), (c) it is unstable and in (d), (e) it is stable. Between scenarios (c) and (d) a Hopf bifurcation occurs and over a very narrow range of \(r\), stable and unstable limit cycles can be observed.]]		[[Image:Well-mixed Ecological PGG - Bifurcations.png\|thumb\|300px\|Bifurcations and dynamical regimes (separated by dashed lines) in heterogeneous populations of cooperators and defectors interacting in ecological public goods games with \(N=8\), \(b=1\), \(d=1.2\) and the multiplication factor \(r\) as the bifurcation parameter. The position of the fixed point \(Q\) changes with increasing \(r\) while the system undergoes a series of different bifurcations: \(Q\) enters on the top left for low \(r\) and leaves at the top right for high \(r\) (transcritical bifurcations). The different dynamical scenarios (a)-(f) apply depending on the location of \(Q\). In (a) and (f) \(Q\) is absent, in (b), (c) it is unstable and in (d), (e) it is stable. Between scenarios (c) and (d) a Hopf bifurcation occurs and over a very narrow range of \(r\), stable and unstable limit cycles can be observed.]]

Hauert at 19:09, 31 March 2012

2012-03-31T19:09:49Z

← Older revision		Revision as of 12:09, 31 March 2012
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	== Population Dynamics ==		== Population Dynamics ==
	In order to combine game dynamics and population dynamics in a replicator equation we assume that \(x\) denotes the density of cooperators, \(y\) the density of defectors and \(z=1-x-y\) the abundance of empty space. Thus, \(x+y\) denotes a normalized population density such that for \(x+y=0\) (or \(z=1\)) the population has gone extinct. The dynamics of \(x, y\) 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 \(z\). This leads to the following population dynamic model:		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 empty 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 x =& x (z (b+f_C)-d)\\		\dot u =& x (w (b+f_C)-d)\\
	\dot y =& y (z (b+f_D)-d)\\		\dot v =& y (w (b+f_D)-d)\\
	\dot z =& -\dot x -\dot y		\dot w =& -\dot u -\dot v
	\end{align*}		\end{align*}

	This system of equations represents a natural extension of the replicator dynamics. If the population density \(x+y\) is kept constant (\(\dot z = 0\)) by adjusting the death rate accordingly, i.e. by setting \(d = z(b+\bar f/(1-z))\), where \(\bar f = x\ f_C+y\ f_D\) denotes the mean fitness, the traditional replicator dynamics is recovered (upon normalizing \(x+y=1\)). The average payoffs \(f_C\) and \(f_D\) are determined by the actual game theoretical interactions under consideration.		This system of equations represents a natural extension of the replicator dynamics. If the population density \(u+v\) is kept constant (\(\dot w = 0\)) by adjusting the death rate accordingly, i.e. by setting \(d = w(b+\bar f/(1-w))\), where \(\bar f = u\ f_C+v\ f_D\) denotes the mean fitness, the traditional replicator dynamics is recovered (upon normalizing \(u+v=1\)). The average payoffs \(f_C\) and \(f_D\) are determined by the actual game theoretical interactions under consideration.

	~~===Payoffs of cooperators and defectors===~~
	~~In a population with a fraction \(x\) cooperators, \(y\) defectors and \(z = 1 - x - y\) available space, then the average payoff to cooperators \(f_C\) and defectors \(f_D\) is given by:~~

	~~\begin{align*}~~
	~~f_D =& r \frac x{1 - z} \left(1 - \frac{1 - z^N}{N (1 - z)}\right)\\~~
	~~f_C =& f_D - F(z)~~
	~~\end{align*}~~
	~~with~~
	~~\begin{align*}~~
	~~F(z) =&\ 1 + (r - 1) z^{N - 1} - r \frac{1 - z^N}{N (1 - z)}.~~
	~~\end{align*}~~

	Note that this motivation assumes that the benefits of the public good is contingent on social interactions, i.e. a single participant in the public goods interaction cannot increase its capital. For a detailed derivation of the formulas please consult the scientific articles in the reference section.

	[[Category:Tutorial]]		[[Category:Tutorial]]

Wikiadmin at 08:44, 31 March 2012

2012-03-31T08:44:09Z

← Older revision		Revision as of 01:44, 31 March 2012
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	In order to combine game dynamics and population dynamics in a replicator equation we assume that \(x\) denotes the density of cooperators, \(y\) the density of defectors and \(z=1-x-y\) the abundance of empty space. Thus, \(x+y\) denotes a normalized population density such that for \(x+y=0\) (or \(z=1\)) the population has gone extinct. The dynamics of \(x, y\) 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 \(z\). This leads to the following population dynamic model:		In order to combine game dynamics and population dynamics in a replicator equation we assume that \(x\) denotes the density of cooperators, \(y\) the density of defectors and \(z=1-x-y\) the abundance of empty space. Thus, \(x+y\) denotes a normalized population density such that for \(x+y=0\) (or \(z=1\)) the population has gone extinct. The dynamics of \(x, y\) 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 \(z\). This leads to the following population dynamic model:

	\[\dot x = x (z (b+f_C)-d)\]		\begin{align*}
	\[\dot y = y (z (b+f_D)-d)\]		\dot x =& x (z (b+f_C)-d)\\
	\[\dot z = -\dot x -\dot y\]		\dot y =& y (z (b+f_D)-d)\\
			\dot z =& -\dot x -\dot y
			\end{align*}

	This system of equations represents a natural extension of the replicator dynamics. If the population density \(x+y\) is kept constant (\(\dot z = 0\)) by adjusting the death rate accordingly, i.e. by setting \(d = z(b+\bar f/(1-z))\), where \(\bar f = x f_C+y f_D\) denotes the mean fitness, the traditional replicator dynamics is recovered (upon normalizing \(x+y=1\)). The average payoffs \(f_C\) and \(f_D\)are determined by the actual game theoretical interactions under consideration.		This system of equations represents a natural extension of the replicator dynamics. If the population density \(x+y\) is kept constant (\(\dot z = 0\)) by adjusting the death rate accordingly, i.e. by setting \(d = z(b+\bar f/(1-z))\), where \(\bar f = x\ f_C+y\ f_D\) denotes the mean fitness, the traditional replicator dynamics is recovered (upon normalizing \(x+y=1\)). The average payoffs \(f_C\) and \(f_D\) are determined by the actual game theoretical interactions under consideration.

	===Payoffs of cooperators and defectors===		===Payoffs of cooperators and defectors===
	In a population with a fraction \(x\) cooperators, \(y\) defectors and \(z = 1 - x - y\) available space, then the average payoff to cooperators \(f_C\) and defectors \(f_D\) is given by:		In a population with a fraction \(x\) cooperators, \(y\) defectors and \(z = 1 - x - y\) available space, then the average payoff to cooperators \(f_C\) and defectors \(f_D\) is given by:

	\[f_D = r \frac x{1 - z} (1 - \frac{1 - z^N}{N (1 - z)})\]		\begin{align*}
	\[f_C = f_D - F(z)\]		f_D =& r \frac x{1 - z} \left(1 - \frac{1 - z^N}{N (1 - z)}\right)\\
			f_C =& f_D - F(z)
			\end{align*}
	with		with
	\[F(z) = 1 + (r - 1) z^{N - 1} - r \frac{1 - z^N}{N (1 - z)}.\]		\begin{align*}
			F(z) =&\ 1 + (r - 1) z^{N - 1} - r \frac{1 - z^N}{N (1 - z)}.
			\end{align*}

	Note that this motivation assumes that the benefits of the public good is contingent on social interactions, i.e. a single participant in the public goods interaction cannot increase its capital. For a detailed derivation of the formulas please consult the scientific articles in the reference section.		Note that this motivation assumes that the benefits of the public good is contingent on social interactions, i.e. a single participant in the public goods interaction cannot increase its capital. For a detailed derivation of the formulas please consult the scientific articles in the reference section.

	[[Category:Tutorial]]		[[Category:Tutorial]]

Wikiadmin: moved Evolutionary Games and Population Dynamics / Well-mixed populations to Evolutionary Games and Population Dynamics/Well-mixed populations

2012-03-31T08:31:31Z

moved Evolutionary Games and Population Dynamics / Well-mixed populations to Evolutionary Games and Population Dynamics/Well-mixed populations

← Older revision	Revision as of 01:31, 31 March 2012
(No difference)

Hauert at 23:49, 21 March 2012

2012-03-21T23:49:37Z

Show changes

Wikiadmin at 22:24, 26 January 2010

2010-01-26T22:24:31Z

← Older revision		Revision as of 15:24, 26 January 2010
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	{{-}}		{{-}}
	== Evolutionary scenarios ==		== Evolutionary scenarios ==
	All of the following examples and suggestions are meant as inspirations for further experimenting with the ''~~VirtualLabs~~''. 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.

	{{Legend:PlainAB}}		{{Legend:PlainAB}}
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	<div class="lab_description EcoPGG">		<div class="lab_description EcoPGG">
	[[Image:Well-mixed Ecological PGG - no Q, extinction.png \|left\|200px]]		[[Image:Well-mixed Ecological PGG - no Q, extinction.png \|left\|200px]]
	==== [[~~VirtualLab~~: Ecological Public Goods Game - No Q, extinction\|(a) No ''Q'', extinction]]====		==== [[EvoLudoLab: Ecological Public Goods Game - No Q, extinction\|(a) No ''Q'', extinction]]====
	No matter what the initial configuration of the cooperators and defectors, the population will invariably go extinct.		No matter what the initial configuration of the cooperators and defectors, the population will invariably go extinct.

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	<div class="lab_description EcoPGG">		<div class="lab_description EcoPGG">
	[[Image:Well-mixed Ecological PGG - unstable node.png \|left\|200px]]		[[Image:Well-mixed Ecological PGG - unstable node.png \|left\|200px]]
	==== [[~~VirtualLab~~: Ecological Public Goods Game - Unstable node\|(b) Unstable node]]====		==== [[EvoLudoLab: Ecological Public Goods Game - Unstable node\|(b) Unstable node]]====
	The presence of the interior fixed point <math>Q</math> does not affect the evolutionary end state of the system - the population keeps going extinct irrespective of the initial conditions.		The presence of the interior fixed point <math>Q</math> does not affect the evolutionary end state of the system - the population keeps going extinct irrespective of the initial conditions.

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	<div class="lab_description EcoPGG">		<div class="lab_description EcoPGG">
	[[Image:Well-mixed Ecological PGG - unstable focus.png \|left\|200px]]		[[Image:Well-mixed Ecological PGG - unstable focus.png \|left\|200px]]
	==== [[~~VirtualLab~~: Ecological Public Goods Game - Unstable focus\|(c) Unstable focus]]====		==== [[EvoLudoLab: Ecological Public Goods Game - Unstable focus\|(c) Unstable focus]]====
	For larger <math>r</math>, the interior fixed point <math>Q</math> turns into an unstable focus and - depending on the initial conditions - the population faces extinction in an oscillatory manner.		For larger <math>r</math>, the interior fixed point <math>Q</math> turns into an unstable focus and - depending on the initial conditions - the population faces extinction in an oscillatory manner.

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	<div class="lab_description EcoPGG">		<div class="lab_description EcoPGG">
	[[Image:Well-mixed Ecological PGG - stable limit cycle.png \|left\|200px]]		[[Image:Well-mixed Ecological PGG - stable limit cycle.png \|left\|200px]]
	==== [[~~VirtualLab~~: Ecological Public Goods Game - Stable limit cycle\|Stable limit cycle - super-critical Hopf bifurcation]]====		==== [[EvoLudoLab: Ecological Public Goods Game - Stable limit cycle\|Stable limit cycle - super-critical Hopf bifurcation]]====
	For slightly higher <math>r</math> the interior fixed point <math>Q</math> 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 <math>r</math> the interior fixed point <math>Q</math> 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.

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	<div class="lab_description EcoPGG">		<div class="lab_description EcoPGG">
	[[Image:Well-mixed Ecological PGG - stable focus.png \|left\|200px]]		[[Image:Well-mixed Ecological PGG - stable focus.png \|left\|200px]]
	==== [[~~VirtualLab~~: Ecological Public Goods Game - Stable focus\|(d) Stable focus]]====		==== [[EvoLudoLab: Ecological Public Goods Game - Stable focus\|(d) Stable focus]]====
	Increasing <math>r</math> further leads to a Hopf bifurcation, the interior fixed point <math>Q</math> 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 <math>r</math> further leads to a Hopf bifurcation, the interior fixed point <math>Q</math> 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.
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	<div class="lab_description EcoPGG">		<div class="lab_description EcoPGG">
	[[Image:Well-mixed Ecological PGG - stable node.png \|left\|200px]]		[[Image:Well-mixed Ecological PGG - stable node.png \|left\|200px]]
	==== [[~~VirtualLab~~: Ecological Public Goods Game - Stable node\|(e) Stable node]]====		==== [[EvoLudoLab: Ecological Public Goods Game - Stable node\|(e) Stable node]]====
	Another increase in <math>r</math> turns the interios fixed point <math>Q</math> 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 <math>r</math> turns the interios fixed point <math>Q</math> 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.
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	[[Image:Well-mixed Ecological PGG - no Q, cooperation.png \|left\|200px]]		[[Image:Well-mixed Ecological PGG - no Q, cooperation.png \|left\|200px]]

	==== [[~~VirtualLab~~: 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 <math>r</math>, the interior fixed point <math>Q</math> disappears and the high density saddle node along <math>f=1</math>, 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 <math>r</math>, the interior fixed point <math>Q</math> disappears and the high density saddle node along <math>f=1</math>, 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.
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	<div class="lab_description EcoPGG">		<div class="lab_description EcoPGG">
	[[Image:Well-mixed Ecological PGG - Bautin.png \|left\|200px]]		[[Image:Well-mixed Ecological PGG - Bautin.png \|left\|200px]]
	==== [[~~VirtualLab~~: Ecological Public Goods Game - Bautin\|Multiple limit cycles - Bautin bifurcation]]====		==== [[EvoLudoLab: Ecological Public Goods Game - Bautin\|Multiple limit cycles - Bautin bifurcation]]====
	In this example, for <math>N=12</math>, a stable and an unstable limit cycle exist on one side of the Hopf bifurcation and another stable limit cycle on the other side.		In this example, for <math>N=12</math>, a stable and an unstable limit cycle exist on one side of the Hopf bifurcation and another stable limit cycle on the other side.

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	<div class="lab_description EcoPGG">		<div class="lab_description EcoPGG">
	[[Image:Well-mixed Ecological PGG - unstable limit cycle.png \|left\|200px]]		[[Image:Well-mixed Ecological PGG - unstable limit cycle.png \|left\|200px]]
	==== [[~~VirtualLab~~: Ecological Public Goods Game - Unstable limit cycle\|Unstable limit cycle - sub-critical Hopf bifurcation]]====		==== [[EvoLudoLab: Ecological Public Goods Game - Unstable limit cycle\|Unstable limit cycle - sub-critical Hopf bifurcation]]====
	In the special case <math>b=0</math> sub-critical Hopf-bifurcations always seem to give rise to a pair of stable and unstable limit cycles (Bautin bifurcation). Apparently only for <math>b>0</math> a simple sub-critical Hopf-bifurcation can be observed.		In the special case <math>b=0</math> sub-critical Hopf-bifurcations always seem to give rise to a pair of stable and unstable limit cycles (Bautin bifurcation). Apparently only for <math>b>0</math> a simple sub-critical Hopf-bifurcation can be observed.
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