Multiple patterns formation for an aggregation/diffusion predator-prey system

Abstract

We investigate existence of stationary solutions to an aggregation/diffusion system of PDEs, modelling a two species predator-prey interaction. In the model this interaction is described by non-local potentials that are mutually proportional by a negative constant −α, with α > 0. Each species is also subject to non-local self-attraction forces together with quadratic diffusion effects. The competition between the aforementioned mechanisms produce a rich asymptotic behavior, namely the formation of steady states that are composed of multiple bumps, i.e. sums of Barenblatt-type profiles. The existence of such stationary states, under some conditions on the positions of the bumps and the proportionality constant α, is showed for small diffusion, by using the functional version of the Implicit Function Theorem. We complement our results with some numerical simulations, that suggest a large variety in the possible strategies the two species use in order to interact each other.

Variations in nonlocal interaction range lead to emergent chase-and-run in heterogeneous populations

“…Likewise, the notion of prey-taxis [37], whereby predators seek to locate themselves near prey, is well-understood from both empirical [38] and mathematical [39] perspectives. Both aspects have been studied mathematically in a model similar to ours but with quadratic rather than linear diffusion [40]. However, in empirical situations, it is less clear whether a combination of predator-avoidance and prey-taxis behaviour ever actually leads to the kind of population-level chase-and-run described here.…”

Nonlocal cross-interaction systems on graphs: Energy landscape and dynamics

“…Also, all summands in both sums of ( 24) are nonnegative, i.e., each sum vanishes if and only if all of its summands vanish. We now assume (24) and argue by contradiction. To this end, we additionally assume that there exist two indices 𝜆 0 , 𝜆 1 ∈ {1, .…”

“…The formation of a rich variety of patterns in this class of systems is widely known, for instance in the context of predator-prey dynamics [19], pigment-cell interactions in zebrafish [34,35], or cell-adhesion and tissue-growth [2,14,32,31]. Additionally, at the continuous level, behavior such as engulfment, mixing, and phase-separation can be found in [23,12,5,4,16,9,7,15,24]. As pointed out earlier, the model proposed in [26] features nonlinear mobilities which typically arise in contexts of saturation or crowding phenomena, i.e., as the configuration becomes increasingly crowded the mobility decreases [8,32,5,6,3,30,33,10].…”

Nonlocal Mechanistic Models in Ecology: Numerical Methods and Parameter Inferences

“…Another version of System (2) was introduced in [38] to model social segregation. While the case when N = 1 has received significant attention [31,[39][40][41], as well as versions for N = 2 [42,43], the model has been utilized to study diverse phenomena such as animal territories [26], predator/prey dynamics [44], and human gangs [45]. Its behavior is further investigated in [36], where the authors explore methods to find equilibrium solutions for multiple groups in two dimensions, a critical aspect in connecting the model with data.…”