Biological modelling
AAU
Dynamics of modified predator-prey models
(with P.E. Kloeden) International Journal of Bifurcation and
Chaos 20(9), 2010, 2657-2669
Besides being structurally unstable, the Lotka-Volterra predator-prey model has another shortcoming due to the invalidity of the principle of mass action when the populations are very small. This leads to extremely large populations recovering from unrealistically small ones. The effects of linear modifications to structurally unstable continuous-time predator-prey models in a (small) neighbourhood of the origin are investigated here. In particular, it is shown that typically either a global attractor or repeller arises depending on the choice of coefficients.
The analysis is based on Poincaré mappings, which allow an explicit representation for the classical Lotka-Volterra equations.
(with M. Rasmussen) in Proceedings of the 14th International
Conference on Difference Equations and Applications, Difference
Equations and Applications, M. Bohner, Z. Dosla, G. Lasas, M. Ünal, A. Zafer, eds., 2009, 253-268
We investigate how chaotic time series generated as orbits of the logistic map can be regularized by assuming continuous growth over short time intervals after fixed instants. In doing so, we obtain a Feigenbaum-like scenario which visualizes a period bisection transition away from chaos to asymptotic stability and finally ODE-like behavior. Several well-known results on population models and unimodal maps with negative Schwarzian derivative are applied.
Christian Pötzsche (Jan 2015)
Modelling the spread of Phytophthora
(with A. Henkel and J. Müller) Journal of Mathematical Biology 65 (6-7), 2012, 1359-1385
We consider a model for the morphology and growth of the fungus-like plant pathogen Phytophthora using the example of Phytophthora plurivora utilizing a correlated random walk describing the density of tips. This random walk incorporates a delay in branching behavior: newly split tips only start to grow after a short while. First we questioned the effect of this delay on the running fronts, for uniform- as well as non-uniform turning kernels. We find that this delay basically influences the slope of the front and therewith the way of spatial appropriation. This theoretical prediction is confirmed by the growth of Phytophthora in experiments performed in Petri dishes.
The second question we are dealing within this paper is concerning the manner tips are interacting, especially the point why tips stop to grow “behind” the interface of the front, respectively in confrontation experiments at the interface between two colonies. The combination of experimental data about the spatial structured time course of the glucose concentration and simulations of a model taking into account both, tips and glucose, reveals that nutrient depletion is most likely the central mechanism of tip interaction and hyphal growth. We presume that this is the growing mechanism of this Phytophthora in infected plant tissue and this the pathogen will sap its hosts via energy depletion and tissue destruction in infected areas.
Bet-hedging in stochastically switching
(with J. Müller, B.A. Hense, T. Fuchs and M. Utz) Journal of Theoretical Biology 336, 2013, 144-167
We investigate a population dynamics model incorporating a stochastically switching environment by means of adaptive dynamics. The aim is to extend known results to the situation at hand, and to deepen the understanding of the range of validity of these results. We find three different types of evolutionarily stable strategies (ESS) in dependence on the frequency at which the environment changes: for rapid change, a monomorphic phenotype, adapted to the mean environment is an ESS. In a middle range, a bimorphic bet-hedging phenotype is optimal, while for slowly changing environments a monomorphic phenotype adapted to the current environment is best. While the last result is only obtained by means of heuristic arguments and simulations, the first two results are based on the analysis of Lyapunov exponents for stochastic switching systems.
Qualitative analysis of a nonautonomous Beverton-Holt Ricker model
(with T. Hüls), SIAM Journal of Applied Dynamical Systems 13 (4), 2014, 1442-1488
We explore a planar discrete-time model from population dynamics subject to a general time-varying environment in order to illustrate the recent theory of nonautonomous dynamical systems. Given such a setting, the mathematical standard tools from classical dynamical systems and bifurcation theory cannot be employed, since for instance equilibria typically do not exist or eigenvalues yield no stability information. For this reason, we apply a combination of contemporary analytical and numerical techniques adequate to tackle such situations.
Nonautonomous bifurcation scenarios in SIR
(with P.E. Kloeden), Mathematical Methods in the Applied Sciences 38, 2015, 3495-3518
The standard obstacles in developing a bifurcation theory for nonautonomous differential equations are the lack of steady state equilibria and the insignificance of eigenvalues in stability investigations. For this reason, various different techniques have been proposed to specify changes in the qualitative behavior of time-dependent dynamical systems. In this paper, we investigate and compare several approaches to nonautonomous bifurcations using SIR-like models from epidemiology as a paradigm. These models are sufficiently simple to allow explicit solutions to a large extent and consequently enable a detailed discussion of the different results.
