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  • Kisdi, Eva; Geritz, Stefan A.H. (2016)
    We study the joint adaptive dynamics of n scalar-valued strategies in ecosystems where n is the maximum number of coexisting strategies permitted by the (generalized) competitive exclusion principle. The adaptive dynamics of such saturated systems exhibits special characteristics, which we first demonstrate in a simple example of a host-pathogen-predator model. The main part of the paper characterizes the adaptive dynamics of saturated polymorphisms in general. In order to investigate convergence stability, we give a new sufficient condition for absolute stability of an arbitrary (not necessarily saturated) polymorphic singularity and show that saturated evolutionarily stable polymorphisms satisfy it. For the case , we also introduce a method to construct different pairwise invasibility plots of the monomorphic population without changing the selection gradients of the saturated dimorphism.
  • Gyllenberg, Mats; Hanski, Ilkka; Lindstrom, Torsten (2017)
    Within the framework of adaptive dynamics we consider the evolution by natural selection of reproductive strategies in which individuals may adjust their reproductive behaviour in response to changing environmental conditions. As a specific example we considered a discrete-time model in which possible fluctuations in the environmental conditions are caused by predator-prey interaction. Our main findings include: (1) Coexistence between two fixed strategies (i.e., strategies that do not adjust to changing environmental conditions) is impossible; there exists a best fixed strategy, which invades and ousts all other fixed strategies. (2) A necessary condition for conditional (adjustable) strategies to evolve is that there are fluctuations in the environmental conditions. Predator-prey interactions may cause such fluctuations and under natural assumptions there exists an optimal conditional strategy which is uninvadable and invades and ousts all other strategies.
  • Kisdi, Eva (2015)
    Evolutionary singularities are central to the adaptive dynamics of evolving traits. The evolutionary singularities are strongly affected by the shape of any trade-off functions a model assumes, yet the trade-off functions are often chosen in an ad hoc manner, which may unjustifiably constrain the evolutionary dynamics exhibited by the model. To avoid this problem, critical function analysis has been used to find a trade-off function that yields a certain evolutionary singularity such as an evolutionary branching point. Here I extend this method to multiple trade-offs parameterized with a scalar strategy. I show that the trade-off functions can be chosen such that an arbitrary point in the viability domain of the trait space is a singularity of an arbitrary type, provided (next to certain non-degeneracy conditions) that the model has at least two environmental feedback variables and at least as many trade-offs as feedback variables. The proof is constructive, i.e., it provides an algorithm to find trade-off functions that yield the desired singularity. I illustrate the construction of trade-offs with an example where the virulence of a pathogen evolves in a small ecosystem of a host, its pathogen, a predator that attacks the host and an alternative prey of the predator.
  • Getto, Philipp; Gyllenberg, Mats; Nakata, Yukihiko; Scarabel, Francesca (2019)
    We consider a mathematical model describing the maturation process of stem cells up to fully mature cells. The model is formulated as a differential equation with state-dependent delay, where maturity is described as a continuous variable. The maturation rate of cells may be regulated by the amount of mature cells and, moreover, it may depend on cell maturity: we investigate how the stability of equilibria is affected by the choice of the maturation rate. We show that the principle of linearised stability holds for this model, and develop some analytical methods for the investigation of characteristic equations for fixed delays. For a general maturation rate we resort to numerical methods and we extend the pseudospectral discretisation technique to approximate the state-dependent delay equation with a system of ordinary differential equations. This is the first application of the technique to nonlinear state-dependent delay equations, and currently the only method available for studying the stability of equilibria by means of established software packages for bifurcation analysis. The numerical method is validated on some cases when the maturation rate is independent of maturity and the model can be reformulated as a fixed-delay equation via a suitable time transformation. We exploit the analytical and numerical methods to investigate the stability boundary in parameter planes. Our study shows some drastic qualitative changes in the stability boundary under assumptions on the model parameters, which may have important biological implications.