Intricate dynamics caused by facilitation in competitive environments within polluted habitat patches (Q2878025)

The paper deals with the existence, multiplicity and stability of positive equilibria of NEWLINE\[CARRIAGE_RETURNNEWLINE\partial_t u-\partial_{xx} u=\lambda u+a_b(x)u^{p}, x \in (0,1), t>0,CARRIAGE_RETURNNEWLINE\]NEWLINE NEWLINE\[CARRIAGE_RETURNNEWLINEu(0,t)=u(1,t)=M, t>0,CARRIAGE_RETURNNEWLINE\]NEWLINE NEWLINE\[CARRIAGE_RETURNNEWLINEu(x,0)=u_0(x), x \in (0,1),CARRIAGE_RETURNNEWLINE\]NEWLINE where \(u_0>0\) is a given positive function, \(M>0,p>1\), \(\lambda<0\) is regarded as a real parameter. The function \(a_b(x)\) is piece-wise constant and equal to \(-c\) on \([0, \alpha)\cup (1-\alpha,1]\) and to \(b\) on \([\alpha, 1-\alpha]\) for some \(\alpha \in(0,0.5), b \geq 0, c>0.\)NEWLINENEWLINEThe steady states of this problem are the positive solutions of the one-dimensional boundary value problem NEWLINE\[CARRIAGE_RETURNNEWLINE-u^{\prime \prime}=\lambda u+a_b(x)u^{p} \; \text{in} \; (0,1),CARRIAGE_RETURNNEWLINE\]NEWLINE NEWLINE\[CARRIAGE_RETURNNEWLINEu(0)=u(1)=M.CARRIAGE_RETURNNEWLINE\]NEWLINE The authors propose and justify a numerical method for the last problem, compute the bifurcation diagrams and discuss the original problem as a model for the evolution of a species \(u\) in an unfavourable environment.