Spatial social dilemmas promote diversity: Difference between revisions

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| colspan="4" |

Dynamics in well-mixed populations with size \(N=10^4\) for favourable cost-to-benefit ratios, \(r=0.1<1/k\), and harsher conditions, \(r=0.3>1/k\), respectively, as well as weak, moderate or strong selection, \(w=1, 10, 100\).

The interactive labs below illustrate the trait distribution over time where the colours indicate the trait density according to:

{{Legend:Gradient|label=Densities|min=Low|max=High|gradient=white,black,yellow,red}}

|-

! well-mixed !!\(w=1\)!!\(w=10\)!!\(w=100\)

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| colspan="4" |

Dynamics in on \(N=100\times100\) square lattices with \(k=4\) neighbours for favourable cost-to-benefit ratios, \(r=0.1<1/k\), and harsher conditions, \(r=0.3>1/k\), respectively, as well as weak, moderate or strong selection, \(w=1, 10, 100\).

The interactive labs below illustrate the trait distribution over time where the colours indicate the trait density according to:

{{Legend:Gradient|label=Densities|min=Low|max=High|gradient=white,black,yellow,red}}

|-

! lattice, \(k=4\) !!\(w=1\)!!\(w=10\)!!\(w=100\)

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|colspan="3"|

Dynamics on a square \(100\times100\) lattice with \(k=4\) neighbours for favourable cost-to-benefit ratios, \(r=0.1<1/k\), and harsher conditions, \(r=0.3>1/k\), respectively, as well as moderate or strong selection, \(w=10, 100\).

The interactive labs below illustrate the trait distribution over time where the colours indicate the trait density according to:

{{Legend:Gradient|label=Densities|min=Low|max=High|gradient=white,black,yellow,red}}

|-

! '''A-C:''' \(r=0.1\), \(w=10\) !! '''D-F:''' \(r=0.3\), \(w=10\) !! '''G-I:''' \(r=0.3\), \(w=100\)

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|colspan="2"|

Dynamics on a square \(100\times100\) lattice with \(k=4\) neighbours for saturating benefits, \(B(x)=b_0\left(1-\exp{(-b_1 x)}\right)\) and linear costs \(C(x)=c_0 x\) for \(b_0=8, b_1=1, c_0=0.7\) with \(b_0=8, b_1=1, c_0=0.7\) with weak, \(w=1\) and stronger \(w=10\) selection.

The interactive labs below illustrate the trait distribution over time where the colours indicate the trait density according to:

{{Legend:Gradient|label=Densities|min=Low|max=High|gradient=white,black,yellow,red}}

|-

! '''A-C:''' weak selection \(w=1\) !! '''D-F:''' stronger selection \(w=10\)

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|}

Interestingly, the adaptive dynamics analysis again misses subtle but intriguing effects arising from strong selection. A pairwise invasibility plot for strong selection is shown in Fig. 3d) and reveals that \(x^\ast\) is susceptible to invasion by mutants with slightly higher investments. In individual-based models, this effectively turns \(x^\ast\) into a (degenerate) branching point, i.e., a starting point for evolutionary diversification. The diversification into coexisting high and low investors has already been observed in [[#References|Killingback et al. 1999]], but the underlying mechanism had not been addressed. The earlier results were based on a different, deterministic update rule, according to which a focal individual imitated the strategy of the best performing neighbour, including itself, but this update rule essentially corresponds to death-birth updating with very strong selection (the only difference being that the focal individual is removed). Hence the diversification reported in [[#References|Killingback et al. 1999]] is of the same type as the one seen here for strong selection.

Interestingly, the adaptive dynamics analysis again misses subtle but intriguing effects arising from strong selection. A pairwise invasibility plot for strong selection is shown in Fig. 3D) and reveals that \(x^\ast\) is susceptible to invasion by mutants with slightly higher investments. In individual-based models, this effectively turns \(x^\ast\) into a (degenerate) branching point, i.e., a starting point for evolutionary diversification. The diversification into coexisting high and low investors has already been observed in [[#References|Killingback et al. 1999]], but the underlying mechanism had not been addressed. The earlier results were based on a different, deterministic update rule, according to which a focal individual imitated the strategy of the best performing neighbour, including itself, but this update rule essentially corresponds to death-birth updating with very strong selection (the only difference being that the focal individual is removed). Hence the diversification reported in [[#References|Killingback et al. 1999]] is of the same type as the one seen here for strong selection.

== Evolution in the continuous snowdrift game ==

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The numerator of Eq. \eqref{eq:xs:csd:dB} is positive if and only if the condition for evolution away from zero, \(b_1>\frac{k}{k+1}c_1\), is satisfied. Similarly, the denominator of Eq. \eqref{eq:xs:csd:dB} is positive if and only if \(2b_2(k+1)<c_2k\) (recall \(b_2, c_2<0\)). It is clear from Eq. \eqref{eq:css:csd:dB} that this is also the condition for convergence stability. Thus, if a singular point, \(x^\ast\), exists, cooperative traits either evolve away from zero and \(x^\ast\) is convergence stable, or the trait cannot evolve away from zero and \(x^\ast\) is a repellor.

We note that if the singular point exists, \(x^\ast\), it is shifted towards smaller investments for a given set of parameters, as compared to well-mixed populations with the same parameters: \(x^\ast = \frac{b_1-c_1}{2c_2-4b_2}\). Furthermore, the condition for convergence stability, \(2b_2(k+1)<c_2k\), is less restrictive than the corresponding condition \(2b_2<c_2\) in the well-mixed case. Finally, from \eq{ess:csd:dB} we see that the condition for evolutionary stability is \(b_2(k+2)(k+1)<c_2 k^2\), which is again less restrictive than the corresponding condition \(b_2<c_2\) in the well-mixed case. Combining the two stability conditions shows that evolutionary branching occurs for \(2b_2(k+1)/k<c_2<b_2(k+1)(k+2)/k^2\). Thus, the analytical approach based on pair approximation suggests that population structures tend to inhibit evolutionary diversification by decreasing the range of parameters for which the singular point is an evolutionary branching point (c.f. Fig. 4a & c).

We note that if the singular point exists, \(x^\ast\), it is shifted towards smaller investments for a given set of parameters, as compared to well-mixed populations with the same parameters: \(x^\ast = \frac{b_1-c_1}{2c_2-4b_2}\). Furthermore, the condition for convergence stability, \(2b_2(k+1)<c_2k\), is less restrictive than the corresponding condition \(2b_2<c_2\) in the well-mixed case. Finally, from \eq{ess:csd:dB} we see that the condition for evolutionary stability is \(b_2(k+2)(k+1)<c_2 k^2\), which is again less restrictive than the corresponding condition \(b_2<c_2\) in the well-mixed case. Combining the two stability conditions shows that evolutionary branching occurs for \(2b_2(k+1)/k<c_2<b_2(k+1)(k+2)/k^2\). Thus, the analytical approach based on pair approximation suggests that population structures tend to inhibit evolutionary diversification by decreasing the range of parameters for which the singular point is an evolutionary branching point (c.f. Fig. 4A & C).

[[Image:Diversification - continuous SD (dB equilibria).jpg|825px]]

'''Figure 4:''' Continuous snowdrift game with quadratic benefit and cost functions, \(B(x)=b_1 x+b_2 x^2\) and \(C(x)=c_1 x+c_2 x^2\). Evolutionary outcomes are shown as a function of the benefit parameter \(b_1\) and cost parameter \(c_2\) with \(b_2=-1/4\) and \(c_1 = 2\). Note that \(b_1<2\) violates the assumption \(B(x)>C(x)\) at least for small \(x\) and hence effectively mimics the characteristics of the prisoner's dilemma.

A analytical predictions based on adaptive dynamics in well-mixed populations and

B results from individual-based simulations for populations with \(N=10^4\) individuals.

C analytical predictions based on spatial adaptive dynamics and complementing individual-based simulations on \(100\times 100\) lattices for

D moderate selection, \(w=10\), and

E strong selection, \(w=100\).

In lattice populations the parameter region admitting singular strategies is shifted to both smaller values of \(b_1\) and of \(c_2\) and the size of the region admitting evolutionary branching is markedly smaller than in well-mixed populations (A, C).

Interestingly, spatial adaptive dynamics predicts branching only for parameters where defection dominates in well-mixed populations, \(b_1<c_1\), mimicking the continuous prisoner's dilemma. For weak to moderate selection predictions by adaptive dynamics (A, C) are in good agreement with results from individual-based simulations (B, D), where equilibrium investment levels range from the minimum (black) to intermediate (grey) and the maximum (white) augmented by convergence stability (red) and evolutionary instability (blue) with the overlapping region indicating evolutionary branching (maroon) in adaptive dynamics and diversification in simulations.

For strong selection (E) striking differences arise with a much increased region of diversification. The points labelled a-d indicate the parameter settings for the invasion analysis in Fig. 5. Note that the automated classification of investment distributions becomes more difficult whenever the singular investment \(x^\ast\) is close to zero or one.

The results for well-mixed populations summarized in Fig. 4A & B can be replicated and verified in the tutorial [[Origin of Cooperators and Defectors]]. The following labs illustrate the dynamical regimes in spatially structured populations.

====Interactive labs, Figure 4====

{| class=wikitable align=center

|+

| colspan="2" |

Dynamical regimes in the spatial continuous snowdrift game for moderate selection, \(w=10\), on \(N=100\times100\) square lattices with \(k=4\) neighbours.

The interactive labs below illustrate the trait distribution over time where the colours indicate the trait density according to:

{{Legend:Gradient|label=Densities|min=Low|max=High|gradient=white,black,yellow,red}}

|-

! branching, \(c_2=-0.5,b_1=1.65\)

|{{EvoLudoTrigger|

options="--game cSD --benefitfcn 11 --benefitparams 1.65,-0.25 --costfcn 1 --costparams 2,-0.5 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 10 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}

|-

! attractor, \(c_2=-0.3,b_1=1.9\)

|{{EvoLudoTrigger|

options="--game cSD --benefitfcn 11 --benefitparams 1.9,-0.25 --costfcn 1 --costparams 2,-0.3 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 10 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}

|-

! repellor, \(c_2=-0.75,b_1=1.5\)

|{{EvoLudoTrigger|

options="--game cSD --benefitfcn 11 --benefitparams 1.5,-0.25 --costfcn 1 --costparams 2,-0.75 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 10 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}

|-

! zero investments, \(c_2=-0.5,b_1=1.5\)

|{{EvoLudoTrigger|

options="--game cSD --benefitfcn 11 --benefitparams 1.5,-0.25 --costfcn 1 --costparams 2,-0.5 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 10 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}

|-

! full investments, \(c_2=-0.5,b_1=2\)

|{{EvoLudoTrigger|

options="--game cSD --benefitfcn 11 --benefitparams 2,-0.25 --costfcn 1 --costparams 2,-0.5 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 10 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}

|-

|}

Strong selection greatly enhance the parameter region resulting in a diversity of traits, see Fig. 4E. The parameter combinations marked with points a,b,c,d refer to modes of diversification that do not rely on evolutionary branching and only occur in spatially structured populations.

[[Image:Diversification - continuous SD (dB modes).jpg|825px]]

'''Figure 5:''' Spatial modes of diversification in the continuous snowdrift game with quadratic benefit and cost functions.

A-D depict pairwise invasibility plots (PIP, top row) for the four scenarios illustrating increased spatial diversification due to strong selection (c.f. parameter combinations a-d marked in Fig. 4). In all cases the width of the region of disadvantageous mutants decreases with selection strength (grey for \(w=1\); black for \(w=100\)).

A the PIP suggests gradual evolution towards minimal investments, except for smaller resident traits, where not only lower investing mutants can invade but also those making markedly higher investing.

B higher investing mutants can always invade, but so can traits investing markedly less.

C selection strength distorts the PIP in the vicinity of the convergence and evolutionarily stable \(x^\ast\) resulting in a degenerate form of branching (c.f. Fig. 3).

D the PIP indicates that \(x^\ast\) is a repellor such that residents with \(x<x^\ast\) are invaded by lower investors while those with \(x>x^\ast\) by higher investors. However, as a consequence of strong selection, mutants with markedly higher (lower) investments can also invade.

E-H depict corresponding plots of regions of mutual invasibility (PIP\(^\text{2}\), white regions). Regions where mutants or residents are unable to invade (grey) are marked with \((+,-)\) and \((-,+)\), respectively. The vector field shows the [[Spatial adaptive dynamics#Evolutionary divergence of coexisting traits|evolutionary divergence]] and indicates the direction of selection for two co-existing residents based on analytical approximations of the spatial invasion dynamics. In all cases divergence drives the traits away from the diagonal and hence preserves diversity.

Parameters: \(b_2=-1/4, c_1=2\), A \(b_1=1.55\), \(c_2=-0.6\); B \(b_1=1.65\), \(c_2=-0.625\); C \(b_1=1.9\), \(c_2=-0.3\); D \(b_1=1.5\), \(c_2=-0.72\).

====Interactive labs, Figure 5====

{| class=wikitable align=center

|+

| colspan="2" |

Modes of diversification in the spatial continuous snowdrift game. The parameter settings refer to points a,b,c,d in Fig. 4E.

The interactive labs below illustrate the trait distribution over time where the colours indicate the trait density according to:

{{Legend:Gradient|label=Densities|min=Low|max=High|gradient=white,black,yellow,red}}

|-

! Figure 5A & E, Figure 4E a, \(c_2=-0.6,b_1=1.55\)

|{{EvoLudoTrigger|

options="--game cSD --benefitfcn 11 --benefitparams 1.55,-0.25 --costfcn 1 --costparams 2,-0.6 --delay 100 --fitnessmap exp --geometry n --initmean 0.9 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 100 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}

|-

! Figure 5B & F, Figure 4E b, \(c_2=-0.625,b_1=1.65\)

|{{EvoLudoTrigger|

options="--game cSD --benefitfcn 11 --benefitparams 1.65,-0.25 --costfcn 1 --costparams 2,-0.625 --delay 100 --fitnessmap exp --geometry n --initmean 0.2 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 100 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}

|-

! Figure 5C & G, Figure 4E c, \(c_2=-0.3,b_1=1.9\)

|{{EvoLudoTrigger|

options="--game cSD --benefitfcn 11 --benefitparams 1.9,-0.25 --costfcn 1 --costparams 2,-0.3 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 100 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}

|-

! Figure 5D & H, Figure 4E d, \(c_2=-0.72,b_1=1.5\)

|{{EvoLudoTrigger|

options="--game cSD --benefitfcn 11 --benefitparams 1.5,-0.25 --costfcn 1 --costparams 2,-0.72 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 100 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}

|-

|}

== Conclusions & Discussion ==

@@ Line 42: / Line 42: @@
 | colspan="4" |
 Dynamics in well-mixed populations with size \(N=10^4\) for favourable cost-to-benefit ratios, \(r=0.1<1/k\), and harsher conditions, \(r=0.3>1/k\), respectively, as well as weak, moderate or strong selection, \(w=1, 10, 100\).
+The interactive labs below illustrate the trait distribution over time where the colours indicate the trait density according to:
+{{Legend:Gradient|label=Densities|min=Low|max=High|gradient=white,black,yellow,red}}
 |-
 ! well-mixed !!\(w=1\)!!\(w=10\)!!\(w=100\)
@@ Line 68: / Line 70: @@
 | colspan="4" |
 Dynamics in on \(N=100\times100\) square lattices with \(k=4\) neighbours for favourable cost-to-benefit ratios, \(r=0.1<1/k\), and harsher conditions, \(r=0.3>1/k\), respectively, as well as weak, moderate or strong selection, \(w=1, 10, 100\).
+The interactive labs below illustrate the trait distribution over time where the colours indicate the trait density according to:
+{{Legend:Gradient|label=Densities|min=Low|max=High|gradient=white,black,yellow,red}}
 |-
 ! lattice, \(k=4\) !!\(w=1\)!!\(w=10\)!!\(w=100\)
@@ Line 105: / Line 109: @@
 |colspan="3"|
 Dynamics on a square \(100\times100\) lattice with \(k=4\) neighbours for favourable cost-to-benefit ratios, \(r=0.1<1/k\), and harsher conditions, \(r=0.3>1/k\), respectively, as well as moderate or strong selection, \(w=10, 100\).
+The interactive labs below illustrate the trait distribution over time where the colours indicate the trait density according to:
+{{Legend:Gradient|label=Densities|min=Low|max=High|gradient=white,black,yellow,red}}
 |-
 ! '''A-C:''' \(r=0.1\), \(w=10\) !! '''D-F:''' \(r=0.3\), \(w=10\) !! '''G-I:''' \(r=0.3\), \(w=100\)
@@ Line 146: / Line 152: @@
 |colspan="2"|
 Dynamics on a square \(100\times100\) lattice with \(k=4\) neighbours for saturating benefits, \(B(x)=b_0\left(1-\exp{(-b_1 x)}\right)\) and linear costs \(C(x)=c_0 x\) for \(b_0=8, b_1=1, c_0=0.7\) with \(b_0=8, b_1=1, c_0=0.7\) with weak, \(w=1\) and stronger \(w=10\) selection.
+The interactive labs below illustrate the trait distribution over time where the colours indicate the trait density according to:
+{{Legend:Gradient|label=Densities|min=Low|max=High|gradient=white,black,yellow,red}}
 |-
 ! '''A-C:''' weak selection \(w=1\) !! '''D-F:''' stronger selection \(w=10\)
@@ Line 155: / Line 163: @@
 |}
-Interestingly, the adaptive dynamics analysis again misses subtle but intriguing effects arising from strong selection. A pairwise invasibility plot for strong selection is shown in Fig. 3d) and reveals that \(x^\ast\) is susceptible to invasion by mutants with slightly higher investments. In individual-based models, this effectively turns \(x^\ast\) into a (degenerate) branching point, i.e., a starting point for evolutionary diversification. The diversification into coexisting high and low investors has already been observed in [[#References|Killingback et al. 1999]], but the underlying mechanism had not been addressed. The earlier results were based on a different, deterministic update rule, according to which a focal individual imitated the strategy of the best performing neighbour, including itself, but this update rule essentially corresponds to death-birth updating with very strong selection (the only difference being that the focal individual is removed). Hence the diversification reported in [[#References|Killingback et al. 1999]] is of the same type as the one seen here for strong selection.
+Interestingly, the adaptive dynamics analysis again misses subtle but intriguing effects arising from strong selection. A pairwise invasibility plot for strong selection is shown in Fig. 3D) and reveals that \(x^\ast\) is susceptible to invasion by mutants with slightly higher investments. In individual-based models, this effectively turns \(x^\ast\) into a (degenerate) branching point, i.e., a starting point for evolutionary diversification. The diversification into coexisting high and low investors has already been observed in [[#References|Killingback et al. 1999]], but the underlying mechanism had not been addressed. The earlier results were based on a different, deterministic update rule, according to which a focal individual imitated the strategy of the best performing neighbour, including itself, but this update rule essentially corresponds to death-birth updating with very strong selection (the only difference being that the focal individual is removed). Hence the diversification reported in [[#References|Killingback et al. 1999]] is of the same type as the one seen here for strong selection.
 == Evolution in the continuous snowdrift game ==
@@ Line 189: / Line 197: @@
 The numerator of Eq. \eqref{eq:xs:csd:dB} is positive if and only if the condition for evolution away from zero, \(b_1>\frac{k}{k+1}c_1\), is satisfied. Similarly, the denominator of Eq. \eqref{eq:xs:csd:dB} is positive if and only if \(2b_2(k+1)<c_2k\) (recall \(b_2, c_2<0\)). It is clear from Eq. \eqref{eq:css:csd:dB} that this is also the condition for convergence stability. Thus, if a  singular point, \(x^\ast\), exists, cooperative traits either evolve away from zero and \(x^\ast\) is convergence stable, or the trait cannot evolve away from zero and \(x^\ast\) is a repellor.
-We note that if the singular point exists, \(x^\ast\), it is shifted towards smaller investments for a given set of parameters, as compared to  well-mixed populations with the same parameters: \(x^\ast = \frac{b_1-c_1}{2c_2-4b_2}\). Furthermore, the condition for convergence stability, \(2b_2(k+1)<c_2k\), is less restrictive than the corresponding condition \(2b_2<c_2\) in the well-mixed case. Finally, from \eq{ess:csd:dB} we see that the condition for evolutionary stability is \(b_2(k+2)(k+1)<c_2 k^2\), which is again less restrictive than the corresponding condition \(b_2<c_2\) in the well-mixed case. Combining the two stability conditions shows that evolutionary branching occurs for \(2b_2(k+1)/k<c_2<b_2(k+1)(k+2)/k^2\). Thus, the analytical approach based on pair approximation suggests that population structures tend to inhibit evolutionary diversification by decreasing the range of parameters for which the singular point is an evolutionary branching point (c.f. Fig. 4a & c).
+We note that if the singular point exists, \(x^\ast\), it is shifted towards smaller investments for a given set of parameters, as compared to  well-mixed populations with the same parameters: \(x^\ast = \frac{b_1-c_1}{2c_2-4b_2}\). Furthermore, the condition for convergence stability, \(2b_2(k+1)<c_2k\), is less restrictive than the corresponding condition \(2b_2<c_2\) in the well-mixed case. Finally, from \eq{ess:csd:dB} we see that the condition for evolutionary stability is \(b_2(k+2)(k+1)<c_2 k^2\), which is again less restrictive than the corresponding condition \(b_2<c_2\) in the well-mixed case. Combining the two stability conditions shows that evolutionary branching occurs for \(2b_2(k+1)/k<c_2<b_2(k+1)(k+2)/k^2\). Thus, the analytical approach based on pair approximation suggests that population structures tend to inhibit evolutionary diversification by decreasing the range of parameters for which the singular point is an evolutionary branching point (c.f. Fig. 4A & C).
+[[Image:Diversification - continuous SD (dB equilibria).jpg|825px]]
+'''Figure 4:'''  Continuous snowdrift game with quadratic benefit and cost functions, \(B(x)=b_1 x+b_2 x^2\) and \(C(x)=c_1 x+c_2 x^2\). Evolutionary outcomes are shown as a function of the benefit parameter \(b_1\) and cost parameter \(c_2\) with \(b_2=-1/4\) and \(c_1 = 2\). Note that \(b_1&lt;2\) violates the assumption \(B(x)>C(x)\) at least for small \(x\) and hence effectively mimics the characteristics of the prisoner's dilemma.
+	<strong>A</strong> analytical predictions based on adaptive dynamics in well-mixed populations and
+	<strong>B</strong> results from individual-based simulations for populations with \(N=10^4\) individuals.
+	<strong>C</strong> analytical predictions based on spatial adaptive dynamics and complementing individual-based simulations on \(100\times 100\) lattices for
+	<strong>D</strong> moderate selection, \(w=10\), and
+	<strong>E</strong> strong selection, \(w=100\).
+In lattice populations the parameter region admitting singular strategies is shifted to both smaller values of \(b_1\) and of \(c_2\) and the size of the region admitting evolutionary branching is markedly smaller than in well-mixed populations (< strong>A, C</strong>).
+	  Interestingly, spatial adaptive dynamics predicts branching only for parameters where defection dominates in well-mixed populations, \(b_1&lt;c_1\), mimicking the continuous prisoner's dilemma. For weak to moderate selection predictions by adaptive dynamics (<strong>A, C</strong>) are in good agreement with results from individual-based simulations (<strong>B, D</strong>), where equilibrium investment levels range from the minimum (black) to intermediate (grey) and the maximum (white) augmented by convergence stability (red) and evolutionary instability (blue) with the overlapping region indicating evolutionary branching (maroon) in adaptive dynamics and diversification in simulations.
+	  For strong selection (<strong>E</strong>) striking differences arise with a much increased region of diversification. The points labelled <strong>a-d</strong> indicate the parameter settings for the invasion analysis in Fig. 5. Note that the automated classification of investment distributions becomes more difficult whenever the singular investment \(x^\ast\) is close to zero or one.
+The results for well-mixed populations summarized in Fig. 4A & B can be replicated and verified in the tutorial [[Origin of Cooperators and Defectors]]. The following labs illustrate the dynamical regimes in spatially structured populations.
+====Interactive labs, Figure 4====
+{| class=wikitable align=center
+|+
+| colspan="2" |
+Dynamical regimes in the spatial continuous snowdrift game for moderate selection, \(w=10\), on \(N=100\times100\) square lattices with \(k=4\) neighbours.
+The interactive labs below illustrate the trait distribution over time where the colours indicate the trait density according to:
+{{Legend:Gradient|label=Densities|min=Low|max=High|gradient=white,black,yellow,red}}
+|-
+! branching, \(c_2=-0.5,b_1=1.65\)
+|{{EvoLudoTrigger|
+options="--game cSD --benefitfcn 11 --benefitparams 1.65,-0.25 --costfcn 1 --costparams 2,-0.5 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 10 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}
+|-
+! attractor, \(c_2=-0.3,b_1=1.9\)
+|{{EvoLudoTrigger|
+options="--game cSD --benefitfcn 11 --benefitparams 1.9,-0.25 --costfcn 1 --costparams 2,-0.3 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 10 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}
+|-
+! repellor, \(c_2=-0.75,b_1=1.5\)
+|{{EvoLudoTrigger|
+options="--game cSD --benefitfcn 11 --benefitparams 1.5,-0.25 --costfcn 1 --costparams 2,-0.75 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 10 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}
+|-
+! zero investments, \(c_2=-0.5,b_1=1.5\)
+|{{EvoLudoTrigger|
+options="--game cSD --benefitfcn 11 --benefitparams 1.5,-0.25 --costfcn 1 --costparams 2,-0.5 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 10 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}
+|-
+! full investments, \(c_2=-0.5,b_1=2\)
+|{{EvoLudoTrigger|
+options="--game cSD --benefitfcn 11 --benefitparams 2,-0.25 --costfcn 1 --costparams 2,-0.5 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 10 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}
+|-
+|}
+Strong selection greatly enhance the parameter region resulting in a diversity of traits, see Fig. 4E. The parameter combinations marked with points <strong>a,b,c,d</strong> refer to modes of diversification that do not rely on evolutionary branching and only occur in spatially structured populations.
+[[Image:Diversification - continuous SD (dB modes).jpg|825px]]
+'''Figure 5:''' Spatial modes of diversification in the continuous snowdrift game with quadratic benefit and cost functions.
+	<strong>A-D</strong>  depict pairwise invasibility plots (<em>PIP</em>, top row) for the four scenarios illustrating increased spatial diversification due to strong selection (c.f. parameter combinations <strong>a-d</strong> marked in Fig. 4). In all cases the width of the region of disadvantageous mutants decreases with selection strength (grey for \(w=1\); black for \(w=100\)).
+	<strong>A</strong> the <em>PIP</em> suggests gradual evolution towards minimal investments, except for smaller resident traits, where not only lower investing mutants can invade but also those making markedly higher investing.
+	<strong>B</strong>  higher investing mutants can always invade, but so can traits investing markedly less.
+	<strong>C</strong>  selection strength distorts the <em>PIP</em> in the vicinity of the convergence and evolutionarily stable \(x^\ast\) resulting in a degenerate form of branching (c.f. Fig. 3).
+	<strong>D</strong> the <em>PIP</em> indicates that \(x^\ast\) is a repellor such that residents with \(x&lt;x^\ast\) are invaded by lower investors while those with \(x&gt;x^\ast\) by higher investors. However, as a consequence of strong selection, mutants with markedly higher (lower) investments can also invade.
+	<strong>E-H</strong> depict corresponding plots of regions of mutual invasibility (<em>PIP\(^\text{2}\)</em>, white regions). Regions where mutants or residents are unable to invade (grey) are marked with \((+,-)\) and \((-,+)\), respectively. The vector field shows the [[Spatial adaptive dynamics#Evolutionary divergence of coexisting traits|evolutionary divergence]] and indicates the direction of selection for two co-existing residents based on analytical approximations of the spatial invasion dynamics. In all cases divergence drives the traits away from the diagonal and hence preserves diversity.<br>
+<em>Parameters:</em> \(b_2=-1/4, c_1=2\), <strong>A</strong> \(b_1=1.55\), \(c_2=-0.6\); <strong>B</strong> \(b_1=1.65\), \(c_2=-0.625\); <strong>C</strong> \(b_1=1.9\), \(c_2=-0.3\); <strong>D</strong> \(b_1=1.5\), \(c_2=-0.72\).
+====Interactive labs, Figure 5====
+{| class=wikitable align=center
+|+
+| colspan="2" |
+Modes of diversification in the spatial continuous snowdrift game. The parameter settings refer to points <strong>a,b,c,d</strong> in Fig. 4E.
+The interactive labs below illustrate the trait distribution over time where the colours indicate the trait density according to:
+{{Legend:Gradient|label=Densities|min=Low|max=High|gradient=white,black,yellow,red}}
+|-
+! Figure 5A & E, Figure 4E a, \(c_2=-0.6,b_1=1.55\)
+|{{EvoLudoTrigger|
+options="--game cSD --benefitfcn 11 --benefitparams 1.55,-0.25 --costfcn 1 --costparams 2,-0.6 --delay 100 --fitnessmap exp --geometry n --initmean 0.9 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 100 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}
+|-
+! Figure 5B & F, Figure 4E b, \(c_2=-0.625,b_1=1.65\)
+|{{EvoLudoTrigger|
+options="--game cSD --benefitfcn 11 --benefitparams 1.65,-0.25 --costfcn 1 --costparams 2,-0.625 --delay 100 --fitnessmap exp --geometry n --initmean 0.2 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 100 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}
+|-
+! Figure 5C & G, Figure 4E c, \(c_2=-0.3,b_1=1.9\)
+|{{EvoLudoTrigger|
+options="--game cSD --benefitfcn 11 --benefitparams 1.9,-0.25 --costfcn 1 --costparams 2,-0.3 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 100 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}
+|-
+! Figure 5D & H, Figure 4E d, \(c_2=-0.72,b_1=1.5\)
+|{{EvoLudoTrigger|
+options="--game cSD --benefitfcn 11 --benefitparams 1.5,-0.25 --costfcn 1 --costparams 2,-0.72 --delay 100 --fitnessmap exp --geometry n --initmean 0.1 --initsdev 0.01 --intertype a --mutation 0.01 --mutationsdev 0.01 --mutationtype g --numinter 1 --popsize 100x --popupdate d --reportfreq 20 --selection 100 --traitmax 1 --traitmin 0 --view Strategies_-_Distribution"}}
+|-
+|}
 == Conclusions & Discussion ==