Evolutionary dynamics through multispecies competition

Abstract

Disruptive selection, emerging from frequency-dependent intraspecific competition can have very exciting evolutionary outcomes. One such outcome is the origin of new species through an evolutionary branching event. Literature on theoretical models investigating the emergence of disruptive selection is vast, with some investigating the sensitivity of the models on assumptions of the competition and carrying capacity functions’ shapes. What is seldom modeled is what happens once the population escapes its effect via increase phenotypic or genotypic variance. The expectation is mixed: disruptive selection could diminish and ultimately disappear or it could still exist leading to further speciation events through multiple evolutionary branching events. Here, we derive the conditions under which disruptive selection drives two subpopulations that originated at a branching point to other points in trait space where each subpopulation again experiences disruptive selection. We show that the general pattern for further branchings require that the competition function to be even narrower than what is required for the first evolutionary branching. However, we also show that the existence of disruptive selection in higher dimensional systems is also sensitive to the shapes of the functions used.

Appendices

Appendix A

Here, we briefly investigate and justify the idea of symmetry to reduce the high dimensional systems with multiple species or branching events to more manageable systems. We exploit conditions that lead to special branching patterns that allow a dimorphism to be described in terms of a single number. We do this by assuming symmetry by which we assume one of the coalition phenotypes is parameterized by the other; thereby, the three-dimensional problem is reduced to two. The mathematical theory of symmetry is vast, and the details are not needed for our purposes (Weyl 1952; Golubitsky et al. 1988; Hamermesh 1989).

For our purposes, it is sufficient to interpret symmetry as referring to a branching pattern in which the dimorphic mutant invaders are equally distant from the resident in phenotype space (that is, $x_{1}^{*}=-x_{2}^{*}=x^{*}$, for some x ^∗ yet to be determined, and x ₃=0). The mathematical point here is that if the K and C functions are symmetric, then finding symmetric solutions is a much easier problem numerically than finding nonsymmetric branching patterns. For the Lotka–Volterra system (1), this can be described as follows: if the conditions K(x)=−K(−x), C(x ₁,x ₂)=C(x ₂,x ₁) and C(0,x)=C(0,−x) are satisfied (this will be so if, for example, K(x) and C(x ₁,x ₂) depend on x through the quantities x ² and (x ₁−x ₂)², respectively), then finding symmetric branching patterns require the solution of only a scalar nonlinear equation rather than solution of a set of coupled nonlinear equations.

Lastly, we would also like to point out that the foregoing methods are easily generalizable to nonsymmetric solutions if the symmetry conditions are not exactly satisfied by the C and K functions. The argument is based on continuity and serves as the basis of an approach for the computation of the branching patterns, either using numerical methods or analytically as a power series in a small parameter. Then, it can be seen that the symmetric solution x ₂=−x ₁ is replaced by a branching pattern $(x_{1},x_{2})=(x_{1}^{*},x_{2}^{*})$ where $x_{j}^{*}$ are nonsymmetric functions that reduce to the symmetric solution under appropriate conditions. All this implies that the conditions obtained are expected to hold for all systems in an open neighborhood of the symmetric system in parameter space. Therefore, the results obtained are not restrictive.

Appendix B

Here, we further simplify the conditions for the first branching given by Eqs. 8, 10, and 11. Successful invasion by the mutant requires that the corresponding eigenvalue for the invader dynamics (or invasion fitness) be greater than zero. Hence, the requirement for successful invasion by the mutant is:

$$ \Lambda(x_{1},x_{2})=K(x_{2})-C(x_{2},x_{1})K(x_{1})>0 $$

We assume that C(x _j ,x _j )=1 for all j and is the maximum from which it follows that $\left .\frac {\partial C(x_{1},x_{2})}{\partial x_{1}}\right |_{x_{1}=x_{2}}=\left .\frac {\partial C(x_{1},x_{2})}{\partial x_{2}}\right |_{x_{2}=x_{1}}=0$. A necessary condition for an evolutionarily singular strategy is (from Eq. 8 which further reduces to Eq. 15):

$$ \left.\frac{\partial\Lambda(x_{1},x_{2})}{\partial x_{2}}\right|_{x_{1}=x_{2}=x^{*}}=K'(x^{*})-\left.\frac{\partial C(x_{2},x_{1})}{\partial x_{2}}\right|_{x_{1}=x_{2}=x^{*}}K(x^{*})=0 $$

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This evolutionarily singular strategy is a fitness minimum if (from Eq. 10, which further reduces to Eq. 16):

$$\begin{array}{@{}rcl@{}} \left.\frac{\partial^{2}\Lambda(x_{1},x_{2})}{\partial {x_{2}^{2}}}\right|_{x_{1}=x_{2}=x^{*}}=&K^{\prime\prime}(x^{*}) -\left.\frac{\partial^{2}C(x_{2},x_{1})}{\partial {x_{2}^{2}}}\right|_{x_{1}=x_{2}=x^{*}}K(x^{*})>0 \\ \end{array} $$

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Convergence stability (from Eq. 11, which further reduces to Eq. 17) is:

$$\begin{array}{@{}rcl@{}} &&{}-\left.\frac{\partial^{2}C(x_{2},x_{1})}{\partial {x_{1}^{2}}}\right|_{x_{1}=x_{2}=x^{*}}K(x^{*})-\left.2\frac{\partial C(x_{2},x_{1})}{\partial x_{1}}\right|_{x_{1}=x_{2}=x^{*}}K'(x^{*})\\ &&{\kern12pt}\left.-C(x_{2},x_{1})\right|_{x_{1}=x_{2}=x^{*}}K^{\prime\prime}(x^{*})\\ \end{array} $$

$$\begin{array}{@{}rcl@{}} {\kern11pt}>K^{\prime\prime}(x^{*})-\left.\frac{\partial^{2}C(x_{2},x_{1})}{\partial {x_{2}^{2}}}\right|_{x_{1}=x_{2}=x^{*}}K(x^{*}) \end{array} $$

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Appendix C

Here, we further simplify the conditions for the second branching given by Eqs. 12, 13, and 14. Only the resident species x ₁ and x ₂ have nonzero populations at the resident equilibrium (n ₃=0). From N=A ⁻¹ K, where N and K are the vectors of population sizes and carrying capacity functions, respectively, and A ⁻¹ is the inverse of the community matrix, the solution for the two residents can be found to be: $n_{1}=\frac {K(x_{1})-C(x_{1},x_{2})K(x_{2})}{1-C(x_{1},x_{2})C(x_{2},x_{1})}$ and $n_{2}=\frac {K(x_{2})-C(x_{2},x_{1})K(x_{1})}{1-C(x_{1},x_{2})C(x_{2},x_{1})}$. The mutant’s fitness measures its ability to invade the existing two species community:

$$\begin{array}{@{}rcl@{}} \Lambda(x_{1},x_{2},x_{3})&=&K(x_{3})-\frac{C(x_{3},x_{1})-C(x_{3},x_{2})C(x_{2},x_{1})}{1-C(x_{1},x_{2})C(x_{2},x_{1})}K(x_{1})\\&& -\frac{C(x_{3},x_{2})-C(x_{3},x_{1})C(x_{1},x_{2})}{1-C(x_{1},x_{2})C(x_{2},x_{1})}K(x_{2})>0 \end{array} $$

due to symmetry properties outlined in Appendix A, this further reduces to the following at x ₁=x and x ₂=−x:

$$ \Lambda(x,-x,x_{3})=K(x_{3})-\frac{C(x_{3},x)+C(x_{3},-x)}{1+C(x,-x)}K(x) $$

Evolutionarily singular strategies (12) are:

$$ \left.\frac{\partial\Lambda(x,-x,x_{3})}{\partial x_{3}}\right|_{x=x_{3}=x^{*}}=K^{\prime}(x^{*})-\frac{\frac{\partial C(x_{3},-x^{*})}{\partial x_{3}}|_{x_{3}=x^{*}}}{1+C(x^{*},-x^{*})}K(x^{*})=0\\ $$

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which reduces to Eq. 18. These strategies are fitness minima (13) if:

$$\begin{array}{@{}rcl@{}} &&{}\left.\frac{\partial^{2}\Lambda(x,-x,x_{3})}{\partial {x_{3}^{2}}}\right|_{x=x_{3}=x^{*}}\\ &&{\kern10pt}=K^{\prime\prime}(x^{*})-\frac{\left.\frac{\partial^{2}C(x_{3},x^{*})}{\partial {x_{3}^{2}}}\right|_{x_{3}=x^{*}}+\left.\frac{\partial^{2}C(x_{3},-x^{*})}{\partial {x_{3}^{2}}}\right|_{x_{3}=x^{*}}}{1+C(x^{*},-x^{*})}K(x^{*})>0 \\ \end{array} $$

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which reduces to Eq. 19. For convergence stability (left-hand side of the inequality in Eq. 14):

$$ \left.\frac{\partial\Lambda(x,-x,x_{3})}{\partial x}\right|_{x_{3}=x=x^{*}}=K(x)\frac{\frac{\partial C(x_{3},-x)}{\partial x}-\frac{\partial C(x_{3},x)}{\partial x}}{1+C(x,-x)} $$

$$\begin{array}{@{}rcl@{}} &&+2K(x)\frac{\partial C(x,-x)}{\partial x}\frac{C(x_{3},x)+C(x_{3},-x)}{(1+C(x,-x))^{2}}\\ &&{\kern6pt}-K'(x)\frac{C(x_{3},x)+C(x_{3},-x)}{1+C(x,-x)}\\ &&\left.\frac{\partial^{2}\Lambda(x,-x,x_{3})}{\partial x^{2}}\right|_{x_{3}=x=x^{*}}=\frac{3C^{*^{\prime\prime}}-C^{o^{\prime\prime}}}{1+C^{*}}K(x^{*})\\&&{\kern6pt}-K^{\prime\prime}(x^{*})-4K(x^{*})\frac{(C^{*'})^{2}}{(1+C^{*})^{2}}+2K'(x^{*})\frac{C^{*'}}{1+C^{*}} \end{array} $$

where $\left .C^{*}=C(x,-x)\right |_{x=x^{*}}$, $C^{*'}=\left .\frac {\partial C(x_{3},-x)}{\partial x_{3}}\right |_{x_{3}=x^{*}}$, $C^{*^{\prime \prime }}=\left .\frac {\partial ^{2}C(x_{3},-x)}{\partial x_{3}}\right |_{x_{3}=x^{*}}$, and $C^{o^{\prime \prime }}=\left .\frac {\partial ^{2}C(x_{3},x)}{\partial x_{3}}\right |_{x_{3}=x^{*}}$. Note that the right-hand side of the inequality in Eq. 14 is given by Eq. 33. Putting the two together:

$$\begin{array}{@{}rcl@{}} &&\frac{3C^{*^{\prime\prime}}-C^{o^{\prime\prime}}}{1+C^{*}}K(x^{*})-K^{\prime\prime}(x^{*})-4K(x^{*})\frac{(C^{*'})^{2}}{(1+C^{*})^{2}}+2K'(x^{*})\frac{C^{*'}}{1+C^{*}}\\ &&-K^{\prime\prime}(x^{*})+\frac{C^{o^{\prime\prime}}+C^{*^{\prime\prime}}}{1+C^{*}}K(x^{*})>0 \end{array} $$

$$ \frac{4C^{*^{\prime\prime}}}{1+C^{*}}K(x^{*})-2K^{\prime\prime}(x^{*})-4K(x^{*})\frac{(C^{*'})^{2}}{(1+C^{*})^{2}}+2K'(x^{*})\frac{C^{*'}}{1+C^{*}}>0 $$

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which is further reduced to Eq. 20 since K ^′(x)=K(x)C ^∗′/(1+C ^∗) from Eq. 32.

Appendix D

Here, we derive conditions of convergence stability when the second derivative of the carrying capacity function equals zero. Convergence stability to a local minimum or a maximum is determined by the fitness gradient near the evolutionarily singular strategy x ^∗ (Eshel 1983):

$$ D(\theta)=\left.\frac{\partial\Lambda(x_{1},x_{2})}{\partial x_{2}}\right|_{x_{2}=x_{1}=x^{*}+\theta} $$

which should be negative for small values of 𝜃 (or positive for small negative values of 𝜃). Taylor expansion of D(𝜃) is:

$$ D(\theta)=D(0)+D'(0)\theta+\frac{1}{2}D^{\prime\prime}(0)\theta^{2}+\frac{1}{3!}D^{\prime\prime\prime}(0)\theta^{3}+\ldots $$

Since x ^∗ is an evolutionarily singular strategy, D(0)=0, and convergence stability condition is satisfied if:

$$ D'(0)=\left.\left(\frac{\partial^{2}\Lambda(x_{1},x_{2})}{\partial x_{1}\partial x_{2}}+\frac{\partial^{2}\Lambda(x_{1},x_{2})}{\partial {x_{2}^{2}}}\right)\right|_{x_{1}=x_{2}=x^{*}+\theta}<0 $$

Note that the above condition results in Eq. 11 when the cross-derivatives are eliminated (Geritz et al. 1998). If the second derivatives are equal to zero, this derivation fails, and convergence stability depends on the leading order nonvanishing terms of D(𝜃). In order for convergence stability to occur, the leading order term must be of odd order and the coefficient must be negative. In that case:

$$\begin{array}{@{}rcl@{}} D^{\prime\prime}(0)&=&\left(\frac{\partial^{3}\Lambda(x_{1},x_{2})}{\partial {x_{1}^{2}}\partial x_{2}}+2\frac{\partial^{3}\Lambda(x_{1},x_{2})}{\partial x_{1}\partial {x_{2}^{2}}}+\left.\frac{\partial^{3}\Lambda(x_{1},x_{2})}{\partial {x_{2}^{3}}}\right)\right|_{x_{1}=x_{2}=x^{*}+\theta}=0 \end{array} $$

at 𝜃=0. And:

$$\begin{array}{@{}rcl@{}} D^{\prime\prime\prime}(0)&=&\left(\frac{\partial^{4}\Lambda(x_{1},x_{2})}{\partial {x_{1}^{3}}\partial x_{2}}+3\frac{\partial^{4}\Lambda(x_{1},x_{2})}{\partial {x_{1}^{2}}\partial {x_{2}^{2}}}+3\frac{\partial^{4}\Lambda(x_{1},x_{2})}{\partial x_{1}\partial {x_{2}^{3}}}\right.\\ &&{\kern12pt}\left.\left.+\frac{\partial^{4}\Lambda(x_{1},x_{2})}{\partial {x_{2}^{4}}}\right)\right|_{x_{1}=x_{2}=x^{*}+\theta}<0 \end{array} $$

These last two conditions are satisfied when the carrying capacity function is quartic.

Appendix E

Here, we derive the convergent stability conditions (20). For the Gaussian competition and carrying capacity functions:

$$\begin{array}{@{}rcl@{}} &&{}\frac{2\text{exp}\left(\frac{-2\left(x^{*}\right)^{2}}{{\sigma_{c}^{2}}}\right)}{1+\text{exp}\left(\frac{-2\left(x^{*}\right)^{2}}{{\sigma_{c}^{2}}}\right)} \left(\frac{1}{{\sigma_{c}^{2}}}+\frac{4\left(x^{*}\right)^{2}}{{\sigma_{c}^{4}}}\right)-\left(\frac{x^{*}}{{\sigma_{c}^{2}}}\text{tan}h\left(\frac{\left(x^{*}\right)^{2}}{{\sigma_{c}^{2}}}\right)-1\right)^{2} \\&&{\kern12pt}-\frac{\left(x^{*}\right)^{2}}{{\sigma_{k}^{4}}}+\frac{1}{{\sigma_{k}^{2}}}>0 \end{array} $$

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Gaussian competition and quartic-carrying capacity functions:

$$\begin{array}{@{}rcl@{}} &&{}\frac{2\text{exp}\left(\frac{-2\left(x^{*}\right)^{2}}{{\sigma_{c}^{2}}}\right)}{1+\text{exp}\left(\frac{-2\left(x^{*}\right)^{2}}{{\sigma_{c}^{2}}}\right)}\left(\frac{1}{{\sigma_{c}^{2}}}+\frac{4\left(x^{*}\right)^{2}}{{\sigma_{c}^{4}}}\right) -\left(\frac{x^{*}}{{\sigma_{c}^{2}}}\text{tan}h\left(\frac{\left(x^{*}\right)^{2}}{{\sigma_{c}^{2}}}\right)-1\right)^{2}\\ &&{\kern24pt}-\frac{4\left(x^{*}\right)^{6}}{{\sigma_{k}^{8}}}+\frac{6\left(x^{*}\right)^{2}}{{\sigma_{k}^{4}}}>0 \end{array} $$

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Gaussian competition and quadratic-carrying capacity functions:

$$\begin{array}{@{}rcl@{}} &&{}\frac{2\text{exp}\left(\frac{-2\left(x^{*}\right)^{2}}{{\sigma_{c}^{2}}}\right)}{1+\text{exp}\left(\frac{-2\left(x^{*}\right)^{2}}{{\sigma_{c}^{2}}}\right)}\left(\frac{1}{{\sigma_{c}^{2}}}+\frac{4\left(x^{*}\right)^{2}}{{\sigma_{c}^{4}}}\right) -\left(\frac{x^{*}}{{\sigma_{c}^{2}}}\text{tan}h\left(\frac{\left(x^{*}\right)^{2}}{{\sigma_{c}^{2}}}\right)-1\right)^{2}\\ &&{\kern24pt}-\frac{2}{1-\left(x^{*}\right)^{2}}>0 \end{array} $$

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Box-like competition and carrying capacity functions:

$$\begin{array}{@{}rcl@{}} &&{}\frac{\frac{\beta_{c}}{2\sigma_{c}\text{tan}h\beta_{c}}\left(\text{tan}h\left(\frac{2x^{*}}{\sigma_{c}}+1\right)\text{sec}h^{2}\left(\frac{2x^{*}}{\sigma_{c}}+1\right)+\text{tan}h\left(\frac{2x^{*}}{\sigma_{c}}-1\right)\text{sec}h^{2}\left(\frac{2x^{*}}{\sigma_{c}}-1\right)\right)}{1+\frac{1}{2\sigma_{c}\text{tan}h\beta_{c}}\left(\text{tan}h\left(\beta_{c}\left(\left(\frac{2x^{*}}{\sigma_{c}}+1\right)\right)\right)-\text{tan}h\left(\beta_{c}\left(\left(\frac{2x^{*}}{\sigma_{c}}-11\right)\right)\right)\right)}\\ &&{\kern3pc}+\left(\frac{\frac{\beta_{c}}{2\sigma_{c}\text{tan}h\beta_{c}}\left(\text{tan}h\left(\frac{2x^{*}}{\sigma_{c}}+1\right)-\text{tan}h\left(\frac{2x^{*}}{\sigma_{c}}-1\right)\right)}{1+\frac{1}{2\sigma_{c}\text{tan}h\beta_{c}}\left(\text{tan}h\left(\beta_{c}\left(\left(\frac{2x^{*}}{\sigma_{c}}+1\right)\right)\right)-\text{tan}h\left(\beta_{c}\left(\left(\frac{2x^{*}}{\sigma_{c}}-11\right)\right)\right)\right)}\right)^{2}\\&& {}-{\beta_{k}^{2}}\left(\text{tan}h\left(\beta_{k}\left(\frac{x^{*}}{\sigma_{k}}+1\right)\right)\text{sec}h^{2}\left(\beta_{k}\left(\frac{x^{*}}{\sigma_{k}}+1\right)\right)-\text{tan}h\left(\beta_{k}\left(\frac{x^{*}}{\sigma_{k}}-1\right)\right)\text{sec}h^{2}\left(\beta_{k}\left(\frac{x^{*}}{\sigma_{k}}-1\right)\right)\right)>0 \end{array} $$

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Appendix F

Here, we give the conditions for evolutionarily stable coalitions (or branching points) with box-like functions (Eqs. 18 and 19):

$$\begin{array}{@{}rcl@{}} &&{}\frac{\beta_{k}\left(\text{sec}h^{2}\left(\beta_{k}\left(\frac{x^{*}}{\sigma_{k}}+1\right)\right)-\text{sec}h^{2}\left(\beta_{k}\left(\frac{x^{*}}{\sigma_{k}}-1\right)\right)\right)}{\sigma_{k}\left(\text{tanh}\left(\beta_{k}\left(\frac{x^{*}}{\sigma_{k}}+1\right)\right)-\text{tan}h\left(\beta_{k}\left(\frac{x^{*}}{\sigma_{k}}-1\right)\right)\right)}\\ &&{\kern3pc}=\frac{\frac{\beta_{c}}{2\sigma_{c}\text{tan}h\beta_{c}}\left(\text{sec}h^{2}\left(\beta_{c}\left(\left(\frac{2x^{*}}{\sigma_{c}}+1\right)\right)\right)-\text{sec}h^{2}\left(\beta_{c}\left(\left(\frac{2x^{*}}{\sigma_{c}}-11\right)\right)\right)\right)}{1+\frac{1}{2\sigma_{c}\text{tan}h\beta_{c}}\left(\text{tan}h\left(\beta_{c}\left(\left(\frac{2x^{*}}{\sigma_{c}}+1\right)\right)\right)-\text{tan}h\left(\beta_{c}\left(\left(\frac{2x^{*}}{\sigma_{c}}-11\right)\right)\right)\right)} \end{array} $$

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$$\begin{array}{@{}rcl@{}} &&{}\frac{-{\beta_{k}^{2}}\left(\text{tan}h\left(\beta_{k}\left(\frac{x^{*}}{\sigma_{k}}+1\right)\right)\text{sec}h^{2}\left(\beta_{k}\left(\frac{x^{*}}{\sigma_{k}}+1\right)\right)-\text{tan}h\left(\beta_{k}\left(\frac{x^{*}}{\sigma_{k}}-1\right)\right)\text{sec}h^{2}\left(\beta_{k}\left(\frac{x^{*}}{\sigma_{k}}-1\right)\right)\right)}{{\sigma_{k}^{2}}\left(\text{tan}h\left(\beta_{k}\left(\frac{x^{*}}{\sigma_{k}}+1\right)\right)-\text{tan}h\left(\beta_{k}\left(\frac{x^{*}}{\sigma_{k}}-1\right)\right)\right)}\\ &&{\kern3pc}>\frac{\frac{-\beta_{c}\text{tan}h\beta^{*}\text{sec}h^{2}\beta^{*}}{\sigma_{c}\text{tan}h\beta_{c}}-\text{tan}h\left(\frac{2x^{*}}{\sigma_{c}}+1\right)\text{sec}h^{2}\left(\frac{2x^{*}}{\sigma_{c}}+1\right)+\text{tan}h\left(\frac{2x^{*}}{\sigma_{c}}-1\right)\text{sec}h^{2}\left(\frac{2x^{*}}{\sigma_{c}}-1\right)}{1+\frac{1}{2\sigma_{c}\text{tan}h\beta_{c}}\left(\text{tan}h\left(\beta_{c}\left(\left(\frac{2x^{*}}{\sigma_{c}}+1\right)\right)\right)-\text{tan}h\left(\beta_{c}\left(\left(\frac{2x^{*}}{\sigma_{c}}-11\right)\right)\right)\right)} \end{array} $$

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