"The evolution of habitat choice facilitates niche expansion"
Matching habitat choice and local adaptation are two key factors that control the distribution and diversification of species. We study their joint evolution in a structured metapopulation model with a continuous distribution of habitats. Habitat choice follows as the outcome of dispersal with non-random immigration, a process always acknowledged yet rarely incorporated into theoretical models. For fixed local adaptation, we find the evolutionarily stable habitat choice as a function linking the probability of settlement to the local environment. When the local adaptation trait co-evolves, the metapopulation can become polymorphic. Our main result shows that coexisting strains with only slightly different local adaptation traits evolve substantially different habitat choice. In turn, different habitat use selects for divergent local adaptations. We thus propose that the joint evolution of habitat choice and local adaptation can facilitate niche expansion via diversification under wide conditions, also when the local adaptation trait evolving alone would attain an ESS restricted to a narrower niche.
University of Helsinki
"Evolution of maturation time in a stage structure prey-predator model"
We study the evolution of predator's maturation time. Longer maturation time resulting in stronger adult predator allows the predator to improve predation ability, reduce handling time and increase survival probability. To understand whether two predators can attain a stable coexistence via evolutionary branching of maturation time, we apply the method of critical function analysis to construct three different types of trade-off functions between maturation time with predator's capture rate, handling time and death rate, respectively. A new method to calculate the invasion fitness is applied in non-equilibrium dynamics. Evolutionary branching of maturation time is possible in the cases of capture rate and handling time. The coexistence of two predators induced by branching is evolutionarily stable. In the case of death rate, the monomorphic predator population can evolve to an evolutionarily stable strategy.
"PDE Models of Multilevel Selection: The Evolution of Cooperation and the Shadow of Individual Selection"
Here we consider a game theoretic model of multilevel selection in which individuals compete based on their payoff and groups also compete based on the average payoff of group members. Our focus is on the Prisoners’ Dilemma: a game in which individuals are best off cheating, while groups of individuals do best when composed of many cooperators. We analyze the dynamics of the two-level replicator dynamics, a nonlocal hyperbolic PDE describing deterministic birth-death dynamics for both individuals and groups. Comparison principles and an invariant property of the tail of the population distribution are used to characterize the threshold level of between-group selection dividing a regime in which the population converges to a delta function at the equilibrium of the within-group dynamics from a regime in which between-group competition facilitates the existence of steady-state densities supporting greater levels of cooperation. In particular, we see that the threshold selection strength and average payoff at steady state depend on a tug-of-war between the individual-level incentive to be a defector in a many-cooperator group and the group-level incentive to have many cooperators over many defectors. We also find that lower-level selection casts a long shadow: if groups are best off with a mix of cooperators and defectors, then there will always be fewer cooperators than optimal at steady state, even in the limit of infinitely strong competition between groups.
University of Helsinki
"Evolution of density-dependent handling times in a predator-prey model."
The competitive exclusion principle states that in a constant population two species competing for the same limited resource cannot coexist. This cannot be generalised to non-constant populations. In particular, it has been shown that two predator species competing for the same niche can coexist if the population exhibits non-equilibrium dynamics such as limit cycles. In addition to the ecological question, there emerges the problem whether the coexistence of different predator types competing for a single prey is evolutionarily robust and attainable. Geritz et al.  used the theory of adaptive dynamics to study the evolution of the handling time in a model with Holling type II functional response. They found that under certain conditions the handling time undergoes evolutionary branching and leads to the establishment of the evolutionarily robust coexistence of two predator types. Essential is the assumption of a trade-off between the handling time and the conversion factor connecting the predator's birth rate to its capture rate. It was found that the predator type with the short handling time dominates in the part of the population cycle where the prey is abundant, while the other type prevails when the prey is rare. Ecologically this makes sense: when the prey density is low, prospects of capturing new prey are diminished and it becomes worthwhile to cling to the prey one has already got despite of its gradually decreasing returns. When the prey is common, however, it is easy to replace the partially spent prey by a new one. In this presentation we derive a Holling type II functional response with handling time that depends on the prey density. The ecological setting raises some considerations on the importance of the mechanistic approach when we look at the functions which model the population dynamics as well as leads to new interesting evolutionary questions. Using the theory of adaptive dynamics, we investigate if at least some level of density dependence is favoured whether or not the population is cycling. A further question is if the density dependence can eliminate the possibility of coexistence of different predator types. This makes sense when a single predator with density dependent handling time dynamically shifts its niche between low and high prey densities depending on the phase of the population cycle in a way that a predator with a fixed handling time cannot.
Technical University of Denmark
"Diel Vertical Migration as a Mean Field Game"
The phenomenon of diel vertical migration is one of the largest daily movements of marine species where animals remain in deep, dark water during daylight hours to avoid visual predation and migrate to upper levels at dusk to feed. The migration of each organism can be rationalized as a trade-off between growth and survival with strategies as spatial distributions of the populations. The dynamics driving vertical migration have broad implications for fluxes through the food-web predator-pray interactions [2, 4]; for vertical transport of carbon from upper to deeper layers (i.e. the so-called 'biological carbon pump') with implications for global climate. Here, we present progress of ongoing work on a framework for expressing diel vertical migration as a 'vertical game' in terms of partial differential equations. In the model setup we consider a population of animals distributed in the water column. It is assumed that each animal in this game moves optimally, seeking regions with high growth rate and small mortality, avoiding regions with high population density. The Nash equilibrium for this mean field game is characterized by a system of partial differential equations, which governs the population distribution and migration velocities of animals. The derived system of PDEs has similarities to equations that appear in the fluid dynamics, specifically the Euler equations for compressible inviscid fluids. For the established framework we derive a discretized system of equations to solve the PDEs governing this game theoretical model and present results of numerical simulations. The discretization process is based on the spectral collocation method which has exponential convergence rate for smooth enough functions compared to finite difference schemes. This allows us to reduce computational complexity of the discrete version of PDEs. Recent results either doesn't take into account cost of movement  or doesn't resolve time continuously . We formulate the equations in continuous time, incorporate costs on excessive movements in our model and illustrate the theory with numerical examples.