Spatiotemporal Patterns in a Diffusive Predator-Prey Model with Prey Social Behavior
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Spatiotemporal Patterns in a Diffusive Predator-Prey Model with Prey Social Behavior Salih Djilali1,2
· Soufiane Bentout 3,2
Received: 15 February 2019 / Accepted: 10 September 2019 © Springer Nature B.V. 2019
Abstract Our aim in this paper is to investigate the behavior of pattern formation for a predator-prey model with social behavior and spatial diffusion. Firstly, we give some solution behavior where the non-existence of a non-constant steady state solution has been proved for some values of the diffusion coefficients. On the other hand, by using the LeraySchauder degree theory the existence of the non-constant steady-state solution has been proved under a suitable conditions on the diffusion coefficients. Keywords Predator-prey model · Social behavior · Hopf bifurcation · Local stability · Self-diffusion
1 Introduction From the past century, the study of the ecological systems using mathematical models became an interesting and important topic. One of the most leading topic in ecology is predator-prey interaction. Its importance can be seen through the big number of the proposed models which describes the interaction between the resource (prey) and the consumer (predator) in different cases. An increasing interest has been noticed in modeling those type of interactions. For discussing some of this achievement we introduce a class of a prototyp-
B S. Djilali
[email protected]; [email protected] S. Bentout [email protected]; [email protected]
1
Faculty of Exact and Computer Sciences, Mathematic Department, Hassiba Benbouali University, Chlef, Algeria
2
Laboratoire d’Analyse Non Linéaire et Mathématiques Appliquées, Tlemcen University, Tlemcen, Algeria
3
Department of Mathematics and Informatics, Center of Belhadj Bouchaib, Ain Temouchent, BP 284 RP, 46000, Algeria
S. Djilali, S. Bentout
ical predator-prey model dA = A a − bA − f (A, B) , dt dB = eB f (A, B) − μ , dt A(0) > 0,
(1.1)
B(0) > 0
where A, B are respective population densities of the resources and consumers. a is the reproduction rate for the prey population. b represents the inner competition rate between the prey on the resources. e stands for the conversion of the captured resource into consumer. eμ is the natural death rate of the predator population. f (A, B) is the behavioral characteristic of the consumers species and known by functional response. This functional is the responsible for modeling different interaction between the prey and the predator which can take different forms for modeling a particular way of interaction resource-consumer. Now discussing some traditional and interesting forms · Holling I type f1 (A, B) = αA (where α is the predation rate). αA · Holling II type f2 (A, B) = 1+αt (where th is average handling time). hA 2
αA · Generalized Holling III type f3 (A, B) = 1+βA+γ (where β, γ are the search coeffiA2 cient of the prey for the predator) αA · Ratio-dependent type f4 (A, B) = B+A . αA · Beddington-DeAngelis type f5 (A, B) = 1+βA+γ , B and more functional su
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