A free boundary problem of a predator–prey model with higher dimension and heterogeneous environment

Spreading and vanishing in the diffusive prey–predator model with a free boundary

“…The main objective of this paper is to study the spreading and vanishing phenomena and long time behavior of solution to (1.2). This work can be regarded as a continuation of our previous papers [23,25,26,28,29] in order to have a better understanding of the dynamics for the above two realistic phenomena (i) and (ii). To compare the present work with our previous papers, here we briefly state the main contents of [23,25,26,28,29].…”

“…This work can be regarded as a continuation of our previous papers [23,25,26,28,29] in order to have a better understanding of the dynamics for the above two realistic phenomena (i) and (ii). To compare the present work with our previous papers, here we briefly state the main contents of [23,25,26,28,29]. In [28], the author and Zhao investigated a free boundary problem of (1.1) with double free boundaries in which the prey lives in the whole space but the predator lives in a region enclosed by free boundaries, and in [29] we generalized the results of [28] to higher dimension and heterogeneous environment case.…”

“…To compare the present work with our previous papers, here we briefly state the main contents of [23,25,26,28,29]. In [28], the author and Zhao investigated a free boundary problem of (1.1) with double free boundaries in which the prey lives in the whole space but the predator lives in a region enclosed by free boundaries, and in [29] we generalized the results of [28] to higher dimension and heterogeneous environment case. For the case that b 3 ¼ 0, and either a 1 ¼ a 2 ¼ 0 or b 1 ¼ b 2 ¼ 0, the problem (1.2) has been studied by the author in [25], and in [23] the free boundary condition h 0 ðtÞ ¼ Àlu x at x ¼ hðtÞ was replaced by h 0 ðtÞ ¼ Àlðu x þ qv x Þ with q > 0.…”

“…In general, the Robin boundary conditions utilized here are more complicated than the Dirichlet and Neumann boundary conditions. The problem (1.2) will exhibit different phenomena from those of [23,25,26,28,29] in the following three aspects:…”

Invasion by an inferior or superior competitor: A diffusive competition model with a free boundary in a heterogeneous environment

“…We remark that the similar Lotka-Volterra type model with free boundary was first introduced by Lin [15], in which a prey-predator model was studied and only the existence results were given. Some other works on preypredator model with free boundary can be referred to [20,22,24] and references cited therein. Moreover, there were still some studies caring about Lotka-Volterra competition model.…”