Chemotactic predator-prey dynamics

Abstract

A discrete chemotactic predator-prey model is proposed in which the prey secrets a diffusing chemical which is sensed by the predator and vice versa. Two dynamical states corresponding to catching and escaping are identified and it is shown that steady hunting is unstable. For the escape process, the predator-prey distance is diffusive for short times but exhibits a transient subdiffusive behavior which scales as a power law t 1/3 with time t and ultimately crosses over to diffusion again. This allows to classify the motility and dynamics of various predatory bacteria and phagocytes. In particular, there is a distinct region in the parameter space where they prove to be infallible predators.PACS numbers: 87.17.Jj,05.10.Gg Phagocytes or predatory microbes are hunting their prey by chemotaxis [1, 2], i.e. they sense the concentration of a chemical which is secreted by the prey and is diffusing through the solution [3]. Typically the predator moves along the steepest gradient of the chemical concentration to ultimately find its ejection source. Likewise the prey (for example another microbe) "smells" a secreted chemical from the advancing predator and tries to escape by moving along in the opposite direction of its maximal gradient. This chemotactically coupled predatorprey systems are relevant for many biological microorganisms. In fact, there are many examples of biological relevance for chemotactically coupled predator-prey microorganisms. To name just a few, common microbial predators and phagocytes are Bdellovibrio [4-6], P. aeruginosa [7], D. discoideum [8,9], lymphocytes [10] and M. xanthus [11,12].Previous theoretical investigations have focussed on spatiotemporal pattern formation in predator-prey colonies [13,14] which are typically described by nonlinear reaction-diffusion equations [15,16]. While the latter approaches involve a coarse-grained continuum modelling, there are much less model studies on individual microorganisms. A discrete swarming model of individual self-propelled particles for bacterial colonies has been proposed by Csirok et al [13]. This was elaborated recently by Romanczuk et al [17] based on a related individual model of . Finally individual autochemotactic models have been studied where the microbe follows its own diffusing secretion [19][20][21][22]. In all of these individual models there is no predator involved, apart from a recent study [23] which addressed a lattice model with no chemicals involved.Here we propose a discrete model which describes both the predator and the prey individually and contains explicitly the diffusion of the two chemicals secreted by the predator and the prey together with the Brownian motion of the latter. The deterministic (fluctuation-free) model is analyzed analytically and by numerical solution which is supplemented by Brownian dynamics computer simulations at finite temperature. Depending on the model parameters and the initial distance between predator and prey, two different dynamical processes are identified which correspond to catching an...

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