Estimation of spreading speeds and travelling waves for the lattice pioneer-climax competition system

Figures & data

Figure 1. Typical fitness functions for system (1(1) $\{\begin{array}{rl}& \frac{\mathrm{d}{u}_j}{\mathrm{d}t}=d_1{\mathcal{D}}_2\left[u_j\right]\left(t\right)+u_j\left(t\right)f\left(c_{11}{u}_j\right(t)+c_{12}{v}_j(t\left)\right)-h_1{u}_j\left(t\right),\\ & \frac{\mathrm{d}{v}_j}{\mathrm{d}t}=d_2{\mathcal{D}}_2\left[v_j\right]\left(t\right)+v_j\left(t\right)g\left(c_{21}{u}_j\right(t)+c_{22}{v}_j(t\left)\right)-h_2{v}_j\left(t\right),t\in {\mathbb{R}}^{+},j\in \mathbb{Z},\end{array}$(1) ).

Figure 2. Nullclines and the structure of equilibria of (2(2) $\{\begin{array}{rl}& \frac{\mathrm{d}{u}_j}{\mathrm{d}t}=d_1{\mathcal{D}}_2\left[u_j\right]\left(t\right)+u_j\left(t\right)f\left(a_1{u}_j\right(t)+v_j(t\left)\right)-h_1{u}_j\left(t\right),\\ & \frac{\mathrm{d}{v}_j}{\mathrm{d}t}=d_2{\mathcal{D}}_2\left[v_j\right]\left(t\right)+v_j\left(t\right)g\left(u_j\right(t)+a_2{v}_j(t\left)\right)-h_2{v}_j\left(t\right),\end{array}$(2) ) with (H1) and (H2).

Figure 3. The construction of functions $\overline{g}\left(w\right)$ and $\underset{\_}{g}\left(w\right)$ under the condition (A1).