An extension of the classification of evolutionarily singular strategies in Adaptive Dynamics

Abstract

The existing classification of evolutionarily singular strategies in Adaptive Dynamics (Geritz et al. in Evol Ecol 12:35–57, 1998; Metz et al. in Stochastic and spatial structures of dynamical systems, pp 183–231, 1996) assumes an invasion exponent that is differentiable twice as a function of both the resident and the invading trait. Motivated by nested models for studying the evolution of infectious diseases, we consider an extended framework in which the selection gradient exists (so the definition of evolutionary singularities extends verbatim), but where the invasion fitness may lack the smoothness necessary for the classification à la Geritz et al. We derive the classification of singular strategies with respect to convergence stability and invadability and determine the condition for the existence of nearby dimorphisms. In addition to ESSs and invadable strategies, we observe what we call one-sided ESSs: singular strategies that are invadable from one side of the singularity but uninvadable from the other. Studying the regions of mutual invadability in the vicinity of a one-sided ESS, we discover that two isoclines spring in a tangent manner from the singular point at the diagonal of the mutual invadability plot. The way in which the isoclines unfold determines whether these one-sided ESSs act as ESSs or as branching points. We present a computable condition that allows one to determine the relative position of the isoclines (and thus dimorphic dynamics) from the dimorphic as well as from the monomorphic invasion exponent and illustrate our findings with an example from evolutionary epidemiology.

Appendix: Deducing the fate of dimorphisms nearby one-sided ESSs from the monomorphic invasion fitness

The aim of this Appendix is to show that the coefficients $k_{13}$ and $k_{23}$ in (34) can be related to the Taylor coefficients of the monomorphic invasion exponent. This implies that the fate of dimorphisms nearby evolutionarily singular strategies can be deduced from the monomorphic invasion fitness.

In the region $u_1 \le v \le u_2$, the dimorphic invasion fitness $s_{u_1, u_2}(v)$ has the form

$$\begin{aligned}&\frac{(u_1-v)(u_2-v)}{u_1-u_2}\Big (k_1^+u_1 + k_2^-u_2 + k_{11}u_1^2 + k_{22}u_2^2 \\&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\, +\,k_{12}u_1 u_2 + k_{13}u_1v + k_{23}u_2v + {\mathrm{h.o.t.}}\Big ). \end{aligned}$$

We write the Taylor expansions of $L_+$ and $L_-$ in (9) up to the second order terms as

$$\begin{aligned} L_+(u,v)&= l_1^+u + l_2^+v + l_{11}^+ u^2 + l_{12}^+uv + l_{22}^+v^2 + \mathrm{h.o.t}, \\ L_-(u,v)&= l_1^-u + l_2^-v + l_{11}^- u^2 + l_{12}^-uv + l_{22}^-v^2 + \mathrm{h.o.t}. \end{aligned}$$

Note that $l_1^+ = a_0 - a_1^+, l_2^+ = a_1^+, l_1^- = a_0 - a_1^-$ and $l_2^- = a_1^-$. Next, we write the neutrality curves $C_1$ and $C_2$ as

$$\begin{aligned} C_1 \sim u_1&= \gamma _1 u_2 + {\mathrm{h.o.t.}}, \nonumber \\ C_2 \sim u_2&= \gamma _2 u_1 + {\mathrm{h.o.t.}} \end{aligned}$$

(67)

with

$$\begin{aligned} \gamma _1&= \frac{a_1^- - a_0 }{a_1^-} \end{aligned}$$

(68a)

$$\begin{aligned} \gamma _2&= \frac{ a_1^+ - a_0 }{a_1^+}. \end{aligned}$$

(68b)

On $C_1$, we have $s_{u_1,u_2}(v) = s_{u_2}(v)$ which, by scaling out the factor $u_2-v$ and comparing coefficients corresponding to terms having, as a function of $v$, the exact factor $v^2$, implies that

$$\begin{aligned} k_{13}u_1 + k_{23}u_2 = l_{22}^- (u_2-u_1) \end{aligned}$$

holds on $C_1$. Similarly, we find that on $C_2$

$$\begin{aligned} k_{13}u_1 + k_{23}u_2 = l_{22}^+ (u_2-u_1) \end{aligned}$$

Using (67) we find that $k_{13}$ and $k_{23}$ satisfy

$$\begin{aligned} k_{13}\gamma _1 + k_{23}&= l_{22}^- (1-\gamma _1) \\ k_{13} + k_{23}\gamma _2&= l_{22}^+ (\gamma _2-1), \end{aligned}$$

and conclude that

$$\begin{aligned} k_{13}&= \frac{l_{22}^+(\gamma _2-1)+l_{22}^-(\gamma _1-1) \gamma _2}{1-\gamma _1\gamma _2} \end{aligned}$$

(69a)

$$\begin{aligned} k_{23}&= \frac{l_{22}^+\gamma _1(1-\gamma _2)+l_{22}^-(1-\gamma _1)}{1-\gamma _1\gamma _2} \end{aligned}$$

(69b)

Using (68a,b) and (69a,b) we can rewrite $\lambda$ in (37) as

$$\begin{aligned} \lambda = \frac{a_0(l_{22}^+ a_1^- - l_{22}^- a_1^+)}{a_1^+ + a_1^- - a_0}. \end{aligned}$$

Taking into account the assumption that $a_0>0$ and $a_1^+ + a_1^- - a_0<0$ we arrive at the conclusion of Proposition 2.