Figure 1
From: Multi-species coexistence in Lotka-Volterra competitive systems with crowding effects
The zero isoclines and the long-term variation density of the classical LV competition equation with the inclusion of crowding effect for the four classical cases. Crowding effects (Solid line: d i = 0.5, equilibrium: E2) enable coexistence in all four cases of classical LV system (Broken line: d i = 0.0, equilibrium: E1). (a) \(({K}_{2} < \frac{{K}_{1}}{{\alpha }_{12}},{K}_{1} > \frac{{K}_{2}}{{\alpha }_{21}})\), (b) \(({K}_{2} > \frac{{K}_{1}}{{\alpha }_{12}},{K}_{1} < \frac{{K}_{2}}{{\alpha }_{21}})\), (c) \(({K}_{2} < \frac{{K}_{1}}{{\alpha }_{12}},{K}_{1} < \frac{{K}_{2}}{{\alpha }_{21}})\), and (d) \(({K}_{2} > \frac{{K}_{1}}{{\alpha }_{12}},{K}_{1} > \frac{{K}_{2}}{{\alpha }_{21}})\). Colors indicate x1(red) and x2 (blue). K i : carrying capacity of species i, and α ij : competition coefficient from species j to species i. Parameters value: b i = 1.0, d i = 0.5, mi0 = 0.1, 0 ≤ δ ≤ 5. (a) α12 = 0.8, α21 = 1.2. (b) α12 = 1.2, α21 = 0.8. (c) α12 = 0.5, α21 = 0.4. (d) α12 = 1.2, α21 = 1.3. Initial density x1 = 0.5, x2 = 0.2. We used Anaconda package of the software Python 3.6 for our simulation analysis22.
