Abstract
The purpose of this work is to study the influence of various local models in the equations of diffusion–advection– reaction on the spatial processes of coexistence of predators and prey under conditions of a nonuniform distribution of the carrying capacity. We consider a system of nonlinear parabolic equations to describe diffusion, taxis, and local interaction of a predator and prey in a one-dimensional habitat. Methods. We carried out the study of the system using the dynamical systems approach and a computational experiment based on the method of lines and a scheme of staggered grids. Results. The behavior of the predator – prey system has been studied for various scenarios of local interaction, taking into account the hyperbolic law of prey growth and the Holling effect with nonuniform carrying capacity. We have established paradoxical scenarios of interaction between prey and predator for several modifications of the trophic function. Stationary and nonstationary solutions are analyzed considering diffusion and directed migration of species. Conclusion. The trophic function that considers the heterogeneity of the resource is proposed, which does not lead to paradoxical dynamics.
Highlights
The purpose of this work is to study the influence of various local models in the equations of diffusion–advection– reaction on the spatial processes of coexistence of predators and prey under conditions of a nonuniform distribution of the carrying capacity
We consider a system of nonlinear parabolic equations to describe diffusion, taxis, and local interaction of a predator and prey in a one-dimensional habitat
We carried out the study of the system using the dynamical systems approach and a computational experiment based on the method of lines and a scheme of staggered grids
Summary
Математическая модель пространственно–временного взаимодействия жертвы с плотностью u(x, t) и хищника с плотностью v(x, t) может быть записана в виде системы уравнений [1,4]. Здесь первое слагаемое характеризует диффузию, а второе слагаемое отвечает за направленную миграцию – таксис, определяемый с помощью функций i, которые могут быть выражены в виде [15, 16]. Функция 1 состоит из трёх частей, которые определяют различные виды направленной миграции: таксис жертвы на ресурс p = p(x) и от мест с избыточным скоплением особей своего вида (−β11u), а также от хищника (−β12v). Первое слагаемое в F1 задаёт рост популяции жертвы, причём функция f (u) имеет вид [3]. Первое слагаемое в функции F2 отвечает за естественную убыль хищника. Положительные коэффициенты b1 и b2 характеризуют соответственно убыль жертвы и прирост хищника в результате их контакта. В уравнениях (1)–(4) все коэффициенты могут быть функциями от x и t, но в данной работе предполагается только пространственная зависимость коэффициентов трофической функции хищника b2 и C, причём эта зависимость соотносится с функцией ресурса жертвы.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
