Travelling waves for a size and space structured model in population dynamics: point to sustained oscillating solution connections (Q616478)

The paper deals with the analysis of travelling wave solutions for a model involving a population of plants, whose density \(u\) depends on the time \(t>0,\) its size \(s\) and the spatial position \(x\). The population has a nonlocal spatial reproduction. The travelling wave problem associated to the above model can be written like a second order infinite delay differential equation of the form \[ \frac{1}{\varepsilon^{2}}u^{\prime\prime}(t) =u(t) -\alpha h\left(\int_{0}^{\infty}\gamma(s) u(t-s) \,ds\right) ,\quad t\in\mathbb{R}, \] \[ u(t) \geq0\;\forall t\in\mathbb{R},\quad \sup _{s\in\mathbb{R}}u(s) <\infty,\quad \lim_{t\rightarrow-\infty }u(t) =0,\quad \liminf_{t\rightarrow\infty}u(t) >0. \] The author shows that, under some appropriate hypotheses, this problem admits sustained oscillating solution connections.