Global attractivity in a partial integrodifference equation (Q2763908)

The author considers the following partial difference equations NEWLINENEWLINENEWLINE(1) \(\Delta_2(m,n)= \Delta^2_1u(m-1,n) -\lambda u(m,n) \sum^\infty_{s=0} \exp\bigl(-\gamma u(m,n-s)\bigr)P(s), n=0,1,\dots, m\in \Omega\), NEWLINENEWLINENEWLINE(2) \(\Delta_1 u(m,n)=0,\;n=0,1,\dots,m \in\partial \Omega\),NEWLINENEWLINENEWLINEwhere: \(\Delta_1u(i,j)- u(i+1,j)- u(i,j)\), \(\Delta_2 u(i,j)=u(i,j+1) -u(i,j)\), \(\Delta^2_1 u(i,j)=\Delta_1 (\Delta_1 u(i,j))\), \(\Omega\) is a bounded domain in the lattice plane with exterior boundary \(\partial\Omega\), \(\lambda\) and \(\gamma \) are positive constants and \(P:Z^+\to (-\infty,\infty)\) is a nondecreasing function and \(\sum^\infty_{t=0} P(t)=:P>0\).NEWLINENEWLINENEWLINEThe existence of solutions is guaranteed for (1) if the initial function \(u(m,\theta)= \varphi(m, \theta)\), \((m,\theta) \in\overline \Omega\times (-\infty,0]\), with \(\varphi (\cdot,0)\in \overline \Omega\). The equation (1) has a unique equilibrium \(N\) which satisfies (4) \(\sum^\infty_{s=0} (\exp(-N^* \gamma)P(s)= \lambda N^*\).NEWLINENEWLINENEWLINEThe author assumes the existence of solutions of (1) for each nonnegative initial function \(\varphi\) and obtains a sufficient condition for the global attractivity of the equilibrium \(N\) of (4).