Eco-evolutionary dynamics, density-dependent dispersal and collective behaviour: implications for salmon metapopulation robustness

The spatial dispersal of individuals plays an important role in the dynamics of populations, and is central to metapopulation theory. Dispersal provides connections within metapopulations, promoting demographic and evolutionary rescue, but may also introduce maladapted individuals, potentially lowering the fitness of recipient populations through introgression of heritable traits. To explore this dual nature of dispersal, we modify a well-established eco-evolutionary model of two locally adapted populations and their associated mean trait values, to examine recruiting salmon populations that are connected by density-dependent dispersal, consistent with collective migratory behaviour that promotes navigation. When the strength of collective behaviour is weak such that straying is effectively constant, we show that a low level of straying is associated with the highest gains in metapopulation robustness and that high straying serves to erode robustness. Moreover, we find that as the strength of collective behaviour increases, metapopulation robustness is enhanced, but this relationship depends on the rate at which individuals stray. Specifically, strong collective behaviour increases the presence of hidden low-density basins of attraction, which may serve to trap disturbed populations, and this is exacerbated by increased habitat heterogeneity. Taken as a whole, our findings suggest that density-dependent straying and collective migratory behaviour may help metapopulations, such as in salmon, thrive in dynamic landscapes. Given the pervasive eco-evolutionary impacts of dispersal on metapopulations, these findings have important ramifications for the conservation of salmon metapopulations facing both natural and anthropogenic contemporary disturbances. This article is part of the theme issue ‘Collective movement ecology’.

We aim to understand how straying between populations may affect phenotypic evolution, and how this evolutionary change may in turn lead to changes in ecological dynamics as they unfold. To do so, we build upon Lande's classic formulation of phenotypic evolution [2]. In what follows, we 1) re-derive Lande's phenotypic evolution equation for one population to introduce the reader to the main concepts and assumptions of the framework used, and, 2) explain how we used this framework for the case of two populations connected by dispersal.
I. Phenotypic evolution from one to two sites Trait evolution in one site To keep track of the evolution of a phenotypic trait x with probability density function p and mean x and variance σ 2 , we need a few basic ingredients. First, we define the mean fitness,W of the population of individuals bearing the trait x as: We also define the mean phenotype in the population as and the mean phenotype after selection (but before reproduction), x w [1], as: Then, we can assess how the change in the phenotype x occurs over time using which can be rewritten as: where h 2 is the heritability of the trait, and the difference x w − x(t) is the classical "selection differential" [1,2]. In other words, 5 is none other that the Breeder's equation.
To understand how x changes over time, we first need to understand how the mean fitness of the population changes with a change in the mean trait value or ∂W ∂x (i.e. the shape of the "adaptive landscape"). Using 1 we have: Using Leibniz rule we can pass the derivative under the integral sign, and assuming p is Gaussian (i.e. of the form 1 ), we calculate the derivative of p with respect to x, which leads to: We can now expand in 8 to get, which making use of equations 1 and 3, becomes ∂W ∂x = W σ 2 x w − x σ 2 W , and can be factored as . Using 5, we can rearrange this expression to obtain an equation that relates the change in the average phenotype from one time step to the next to the amount of heritable variation, σ 2 h 2 , and the adaptive landscape ∂W ∂x : Equation 10 (Lande 1976's equation 7), is a staple of evolutionary biology. It is also possible to use 10 and 4 to write a recurrence relationship that allows to simulating evolutionary change over time in a trait x as: Equation 11 thus assumes the existence of one population with a normally distributed trait in which selection occurs before reproduction, the probability of surviving to reproduce depends on the value of the trait different individuals have, and that the trait is normally distributed after reproduction.
Expanding the single site approach to two sites linked by dispersal In what follows we show how Lande's approach can be used to calculate the magnitude of the phenotypic change between two populations with one trait with two different means in each site. We will then 1) show why it is not necessarily useful to use the obtained expression in a simulation context, and 2) introduce an approximation that retains the features important for exploring how trait and population dynamics impact each other in the case of dispersing populations.
We explore the case of two populations that exchange migrants, with a normally distributed trait x that controls recruitment (as explained in the main text), with each population having a different mean, µ i and µ j , and same variance σ 2 . For simplicity, we will focus on a local population i relative to the connected population j.
When individuals from population j stray into population i, and individuals of population i stray into population j, the phenotypic distribution of trait x in population i is a mixture distribution, p(x), of the form: where g(·) is the Gaussian probability density function, and ω i is the proportion of the mixed population that is composed of individuals local to site i relative to those that have dispersed into site i from site j. This distribution has a mean x = ω i µ i +(1−ω i )µ j . As before, we define the mean fitness as in equation 1, only that now p is a mixture distribution. Let us use Lande's approach to obtain an expression for the change in the mean phenotype of the mixture distribution from one time step to the next. We thus ask, what is the change in W with a change in x? We can use Lande's approach [2] to derive equation 7, replacing p(x) by the mixture (equation 12) to obtain To obtain the derivative of equation 13, we introduce a change in variables, which is provided by the fact that x = ω i µ i + (1 − ω i )µ j : We can thus replace equation 14 into 13, which, after expanding and taking the derivatives with respect to x becomes: To simplify notation, at this point we can come back to the original variables µ i and µ j . We then expand and collect terms to get: Using equations 1 and 3, we can define the following quantities, then replace them into equation 17: Using equation 5 and rearranging terms we get: relates how changes in either µ i or µ j lead to changes in fitness, and vice-versa. However, it doesn't tell us anything as to how straying may lead to evolutionary change, or how that may feedback onto ecological dynamics. Because of this, we will once again use the change of variables in 14, to rewrite 23 as: which can be simplified and rearranged to obtain an expression for the change in the mean phenotype of the mixture distribution, By multiplying 25 by W W , we can further rewrite the equation as: which has a similar form to Lande's equation (10), but shows the explicit dependence of the change in the mean of the mix on the proportion of straying individuals. A symmetrical expression can be derived for the other site. While some understanding can be gained using equation 26, the expression does not easily allow us to simulate the change in x over time because the change in µ i and µ j is difficult to track in closed form. Moreover, this equation only holds for the first time step, and as new dispersing individuals arrive and the trait distribution is again updated by the influx of individuals from the other site, the mix becomes a mix of mixes, such that the analytical expressions presented above no longer hold.
The Gaussian approximation used in the main text The above issue presents us with a tradeoff: we can either track changes in phenotype exactly for a single generation, or we can track changes in phenotype across multiple generations using an approximation. Given that our central goal is to gain an understanding of the feedback between evolutionary and ecological processes in salmon populations with dispersal in explicit space, we simplify the complexities created by phenotypic evolution by assuming that, as new migrants arrive, the mixture distribution can be locally approximated by a Gaussian distribution with a mean equal to that of the mix. Based on this assumption, we can directly use Lande's equation (here equation 10), without violating the original assumptions that led to its current form.
The model used in the main text thus assumes the existence of a normally distributed trait in each population with means µ i and µ j . After straying occurs, the populations now have a mixture distribution as in 12. To keep track of the mean of that mix, we approximate the mixture distribution as a Gaussian with mean x = ω i µ i + (1 − ω i )µ j . Selection and reproduction thus occur at the same time leading to a new generation with a normally distributed trait distribution. In a previous paper by Ronce and Kirkpatrick [3], it was shown that assumptions similar to those made here do not qualitatively alter the observed ecological or evolutionary dynamics. As the asymmetry among populations between sites increases (α > 0), vital rates diverge, such that the maximal growth at sites 1 and 2 is now rmax,1 = (1 + α)rmax,2 and β1 = (1 + α)β2 where α is increased from 0 to 0.1. Steady-states for populations with increasingly asymmetric values are shown in warmer (red) tones. (b) Steady states for populations with higher asymmetric values (0.1 ≤ α ≤ 0.4) for vital rates rmax and β. Note the change in scale compared to panel (a). If asymmetry in vital rates is very high, the pitchfork bifurcation occurs at values of m < 0. We observe that increasing the asymmetry in vital rates -even to very high levelsdoes not impact the qualitative nature of the dynamics.