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Register a new userExistence theory and qualitative analysis for a fully cross-diffusive predator-prey system
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This manuscript considers a Neumann initial-boundary value problem for the predator-prey system $$ left{ begin{array}{l} u_t = D_1 u_{xx} - chi_1 (uv_x)_x + u(lambda_1-u+a_1 v), [1mm] v_t = D_2 v_{xx} + chi_2 (vu_x)_x + v(lambda_2-v-a_2 u), end{array} right. qquad qquad (star) $$ in an open bounded interval $Omega$ as the spatial domain, where for $iin{1,2}$ the parameters $D_i, a_i, lambda_i$ and $chi_i$ are positive. Due to the simultaneous appearance of two mutually interacting taxis-type cross-diffusive mechanisms, one of which even being attractive, it seems unclear how far a solution theory can be built upon classical results on parabolic evolution problems. In order to nevertheless create an analytical setup capable of providing global existence results as well as detailed information on qualitative behavior, this work pursues a strategy via parabolic regularization, in the course of which ($star$) is approximated by means of certain fourth-order problems involving degenerate diffusion operators of thin film type. During the design thereof, a major challenge is related to the ambition to retain consistency with some fundamental entropy-like structures formally associated with ($star$); in particular, this will motivate the construction of an approximation scheme including two free parameters which will finally be fixed in different ways, depending on the size of $lambda_2$ relative to $a_2 lambda_1$.
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We construct exact solutions for a system of two nonlinear partial differential equations describing the spatio-temporal dynamics of a predator-prey system where the prey per capita growth rate is subject to the Allee effect. Using the $big(frac{G}{G}big)$ expansion method, we derive exact solutions to this model for two different wave speeds. For each wave velocity we report three different forms of solutions. We also discuss the biological relevance of the solutions obtained.
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