Evolutionarily Stable Strategies for Fecundity and Swimming Speed of Fish

Abstract

Many pelagic fish species have a life history that involves producing a large number of small eggs. This is the result of a trade-off between fecundity and larval survival probability. There are also trade-offs involving other traits, such as larval swimming speed. Swimming faster increases the average food encounter rate but also increases the metabolic cost. Here we introduce an evolutionary model comprising fecundity and swimming speed as heritable traits. We show that there can be two evolutionary stable strategies. In environments where there is little noise in the food encounter rate, the stable strategy is a low-fecundity strategy with a swimming speed that minimises the mean time taken to reach reproductive maturity. However, in noisy environments, for example where the prey distribution is patchy or the water is turbulent, strategies that optimise mean outcomes are often outperformed by strategies that increase inter-individual variance. We show that, when larval growth rates are unpredictable, a high-fecundity strategy is evolutionarily stable. In a population following this strategy, the swimming speed is higher than would be anticipated by maximising the mean growth rate.

Appendix

We define dimensionless variables

$$\begin{aligned} \hat{X} = \frac{X}{x_m}, \qquad \hat{t} = \frac{t}{t_\mathrm {ref}},\qquad \hat{v} = \frac{v}{v_\mathrm {det}}, \end{aligned}$$

where $t_\mathrm {ref}=4cx_m/(b^2x_p^2)$ and $v_\mathrm {det}= bx_p/(2c)$, which is the swimming speed that maximises the expected growth rate, i.e. the deterministic optimum. Then Eq. (1) becomes

$$\begin{aligned} \mathrm{d}\hat{X} = \hat{r}\mathrm{d}\hat{t} + \hat{\sigma }\mathrm{d}\hat{W}, \end{aligned}$$

where

$$\begin{aligned} \hat{r}= & {} \frac{4ac}{b^2 x_p} + 2\hat{v} -\hat{v}^2, \\ \hat{\sigma }^2= & {} S \frac{x_p}{x_m} \left( \frac{4ac}{b^2x_p}+2\hat{v}\right) \end{aligned}$$

and $\hat{W} = W/t_\mathrm {ref}^{1/2}$ is a standard Brownian motion with respect to dimensionless time $\hat{t}$. The initial condition in the new variables is $\hat{X}(0)=x_0/x_m$, and the fish reaches maturity when $\hat{X}=1$. Dropping the hats, this is equivalent to Eq. (1) with the growth rate and diffusivity given in Eqs. (2) and (3).