How Should We Define Fitness in Structured Metapopulation Models? Including an Application to the Calculation of Evolutionarily Stable Dispersal Strategies

We define a fitness concept applicable to structured metapopulations consisting of infinitely many equally coupled patches, and provide means for calculating its numerical value. In addition we introduce a more easily calculated quantity RM that relates to fitness in the same manner as RO relates to fitness in ordinary population dynamics: RM of a mutant is only defined when the resident population dynamics converges to an equilibrium, and RM is larger (smaller) than one if and only if mutant fitness is positive (negative). RM corresponds to the average number of newborn dispersers resulting from the (on average less than one) local colony founded by a newborn disperser. As an example of the usefulness of these concepts we calculate the ES conditional dispersal strategy for individuals that can account for the local population density in their dispersal decisions

Asynchronous exponential growth of semigroups of nonlinear operators

AbstractThe property of asynchronous exponential growth is analyzed for the abstract nonlinear differential equation z′(t) = Az(t) + F(z(t)), t ⩾ 0, z(0) = x ϵ X, where A is the infinitesimal generator of a semigroup of linear operators in the Banach space X and F is a nonlinear operator in X. Asynchronous exponential growth means that the nonlinear semigroup S(t), t ⩾ 0 associated with this problem has the property that there exists λ > 0 and a nonlinear operator Q in X such that the range of Q lies in a one-dimensional subspace of X and limt → ∞ e−λtS(t)x = Qx for all x ϵ X. It is proved that if the linear semigroup generated by A has asynchronous exponential growth and F satisfies ∥F(x)∥ ⩽ f(∥x∥) ∥x∥, where f: R+ → R+ and ∝∞(f(r)r) dr < ∞, then the nonlinear semigroup S(t), t ⩾ 0 has asynchronous exponential growth. The method of proof is a linearization about infinity. Examples from structured population dynamics are given to illustrate the results

Evolutionary Suicide and Evolution of Dispersal in Structured Metapopulations

In this article we study the evolution of dispersal in a structured metapopulation model. The metapopulation consists of a large (infinite)number of local populations living in patches of habitable environment. Dispersal between patches is modeled by a disperser pool and individuals in transit between patches are exposed to a risk of mortality. Occasionally, local catastrophes eradicate a local population: all individuals in the affected patch die, yet the patch remains habitable. The rate at which such disasters occur can depend on the local population size of a patch. We prove that, in the absence of catastrophes, the strategy not to migrate is evolutionarily stable. It is also convergence stable unless there is no mortality during dispersal. Under a given set of environmental conditions, a metapopulation may be viable and yet selection may favor dispersal rates that drive the metapopulation to extinction. This phenomenon is known as evolutionary suicide. We show that in our model evolutionary suicide can occur for certain types of size-dependent catastrophes. Evolutionary suicide can also happen for constant catastrophe rates, if local growth within patches shows an Allee effect. We study the evolutionary bifurcation towards evolutionary suicide and show that a discontinuous transition to extinction is a necessary condition for evolutionary suicide to occur. In other words, if population size smoothly approaches zero at a boundary of viability in parameter space, this boundary is evolutionarily repelling and no suicide can occur

Adaptive correlations between seed size and germination time

We present a model for the coevolution of seed size and germination time within a season when both affect the ability of the seedlings to compete for space. We show that even in the absence of a morphological or physiological constraint between the two traits, a correlation between seed size and germination time is nevertheless likely to evolve. This raises the more general question to what extent a correlation between any two traits should be considered as an a priori constraint or as an evolved means (or instrument) to actually implement a beneficial combination of traits. We derive sufficient conditions for the existence of a positive or a negative correlation. We develop a toy model for seed and seedling survival and seedling growth and use this to illustrate in practice how to determine correlations between seed size and germination time.Peer reviewe

On the Formulation and Analysis of General Deterministic Structured Population Models. I. Linear Theory

We define a linear physiologically structured population model by two rules, one for reproduction and one for "movement" and survival. We use these ingredients to give a constructive definition of next-population-state operators. For the autonomous case we define the basic reproduction ratio Ro and the Malthusian parameter r and we compute the resolvent in terms of the Laplace transform of the ingredients. A key feature of our approach is that unbounded operators are avoided throughout. This will facilitate the treatment of nonlinear models as a next step

On the Formulation and Analysis of General Deterministic Structured Population Models

We define a linear physiologically structured population model by two rules, one for reproduction and one for "movement" and survival. We use these ingredients to give a constructive definition of next-population-state operators. For the autonomous case we define the basic reproduction ratio R0 and the Malthusian parameter r and we compute the resolvent in terms of the Laplace transform of the ingredients. A key feature of our approach is that unbounded operators are avoided throughout. This will facilitate the treatment of nonlinear models as a next step