Propagation Phenomena for a Discrete Diffusive Predator–Prey Model in a Shifting Habitat

Spatial propagation of a lattice predation–competition system with one predator and two preys in shifting habitats

“…The obtained results indicate that in the shifting habitat, the predator-prey model (1) with a pair of competing preys and one predator admits richer propagation phenomena than the associated model with two competing predators and one prey studied in Ref. 27.…”

“…By applying the similar method to that for Ref. [27, Lemma 2.2], we can obtain that $$\mathcal {P}:\Lambda \rightarrow \Lambda$$ is completely continuous with respect to the norm $$|\cdot |_\mu$$. With the Schauder's fixed‐point theorem, we get that $$\mathcal {P}$$ has a fixed point in Λ.…”

“…At the end of this section, we introduce the following result coming from Ref. [27, Proposition 2.4], which will be used in the proof of Theorem 4. Lemma [ 27 , Proposition 2.4] For any $$c>0$$, the discrete diffusive boundary problem $$$$\begin{equation} {\begin{cases} -c\phi ^{\prime }(z)=d\mathcal {D}_2[\phi ](z)+r\phi (z)[\alpha (z)-\phi (z)], \quad z\in \mathbb {R},\\ \phi (-\infty )=0, \quad \phi (\infty )=\alpha (\infty ) \end{cases}} \end{equation}$$$$admits a positive C 1 solution ϕ such that $$\alpha (\infty )\ge \phi (z)\ge \alpha (\infty )-Be^{-\lambda _0z}$$ for all z in $$\mathbb {R}$$, for some $$\lambda _0>0$$ and $$B>0$$, where parameters c , d , and r are positive constants, and $$\alpha (z)$$ satisfies (A1)–(A2) but its limit at ∞ is not assumed to be equal to 1 in general.…”

“…When describing the aggregated dispersion, the discrete diffusive equations are more suitable than continuous reaction-diffusion models. 29 Very recently, Wang and Zhu 27 studied a discrete diffusive predator-prey system involving "two weak competing predators and one prey" in a shifting habitat:…”

“…When describing the aggregated dispersion, the discrete diffusive equations are more suitable than continuous reaction–diffusion models 29 . Very recently, Wang and Zhu 27 studied a discrete diffusive predator–prey system involving “two weak competing predators and one prey” in a shifting habitat: …”