Abstract
In Schoener’s model of intraguild-predation a prey–predator interaction is mixed with the competition of the prey and the predator for food resource supplied to a system with a constant rate. In this work the model is extended to examine the impact of indirect prey taxis which counts for the movement of predator towards the odor released by prey rather than directly towards gradient of prey density (prey taxis) and indirect predator taxis which refers to prey movement opposite to the gradient of a chemical released by predator. The constant coexistence steady state in the model was shown earlier to be globally stable when Schoener’s O.D.E. model is generalized to reaction–diffusion or even prey taxis system. Existence of global-in-time solutions to Schoener’s model with indirect prey taxis is proved for any space dimension while for the case of indirect predator taxis only in 1D. This study reveals that sufficiently large value of taxis sensitivity parameter disturbs the stability of the coexistence steady state giving rise to pattern formation governed by the Hopf bifurcation. Numerical simulations illustrate emergence of taxis driven spatio-temporal periodic patterns.
Highlights
We consider Schoener’s type predator–prey model [1] describing so called intraguild predation in which both predator and prey exploit competitively a common food resource which is available at some constant rate and shared between the predator and the prey
In this work Schoener’s model is extended to study the impact of indirect prey taxis which counts for indirect movement of predator towards the odor released by prey rather than directly towards gradient
Applied Mathematics Letters 125 (2022) 107745 of prey as well as predator taxis which refers to the movement of prey in the opposite direction to the gradient of chemical released by predator (c.f. [5,6,7,8])
Summary
We consider Schoener’s type predator–prey model [1] describing so called intraguild predation in which both predator and prey exploit competitively a common food resource which is available at some constant rate and shared between the predator and the prey. It turns out that proving existence of global in time solutions to each of the problems demands substantially different arguments and leads to restriction of space dimension to n = 1 for the case of Schoener’s model with indirect predator taxis. Setting in (8)f = αwN with q > n and p = ∞ we infer that supt∈[τ ,Tmax){∥∇W (t)∥∞} < ∞ and adjusting the Moser–Alikakos iteration for the P-equation in much the same way as in the proof of [6, Theorem 1.1] we obtain that supt∈[0 ,Tmax){∥P (t)∥∞} < C1 where C1 depends on the uniform L1-bound in (7) It follows that Tmax = +∞ for problem (1)–(3) with (4). On multiplying the P-equation by P for problem (1)–(3) with (5) we use first the Holder inequality and the Gagliardo–Nirenberg inequality (12) to obtain
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