The Gierer-Meinhardt system in the entire space with non-local proliferation rates

TL;DR

This paper introduces a stationary Gierer-Meinhardt system with non-local proliferation rates, analyzing existence and non-existence of positive solutions in the entire space, relevant to biological and ecological modeling.

Contribution

It presents the first analysis of a Gierer-Meinhardt system with non-local proliferation terms, establishing conditions for solutions' existence and non-existence.

Findings

Existence of positive solutions under certain kernel integrability conditions.

Non-existence results when conditions on parameters are not met.

Highlights the impact of non-local terms on system behavior.

Abstract

In this work, we present a novel stationary Gierer-Meinhardt system incorporating non-local proliferation rates, defined as follows: $$ \begin{cases} \displaystyle -\Delta u+\lambda u=\frac{J*u^p}{v^q}+\rho(x) &\quad\mbox{ in }\mathbb{R}^N\, , N\geq 1,\\[0.1in] \displaystyle -\Delta v+\mu v=\frac{J*u^m}{v^s} &\quad\mbox{ in }\mathbb{R}^N.\\[0.1in] \end{cases} $$ This system emerges in various contexts, such as biological morphogenesis, where two interacting chemicals, identified as an activator and an inhibitor, are described, and in ecological systems modelling the interaction between two species, classified as specialists and generalists. The non-local interspecies interactions are represented by the terms $J*u^p, J*u^m$ where the $*$-symbol denotes the convolution operation in $\mathbb{R}^N$ with a kernel $J\in C^1(\mathbb{R}^N\setminus\{0\})$. In the system, we assume that $0<\rho\in C^{0, \gamma}(\mathbb{R}^N)$ with $\gamma\in (0,1)$, while the parameters satisfy $\lambda, \mu, q,m,s>0$ and $p>1$. Under various integrability conditions on the kernel $J$, we establish the existence and non-existence of classical positive solutions in the function space $C^{2, \delta}_{loc}(\mathbb{R}^N).$ These results further highlight the influence of the non-local terms, particularly the proliferation rates, in the proposed model.