Transition of interaction outcomes in a facilitation-competition system of two species

Figure 1.  Phase-plane diagrams of (4) with $d_1\ge 1$. Red and blue lines are the isoclines for $x$ and $y$, respectively. Grey arrows display the direction and strength of the vector fields in the phase-plane space. Fix $r_1=r_2=1, d_1=1, d_2=0.1$ and let the facilitation $c$ vary. (a-b) When $c(=6, 10)$ is small, species $X$ goes to extinction and $Y$ approaches its carrying capacity. (c) When $c(=15)$ is large, the species coexist. (d) When $c(=90)$ is very large, the species coexist and species $Y$ approaches a density extremely less than its carrying capacity

Figure 2.  Phase-plane diagrams of (4) with $d_1< 1$. Red and blue lines are the isoclines for $x$ and $y$, respectively. Grey arrows display the direction and strength of the vector fields in the phase-plane space. Fix $r_1=r_2=1, d_2=0.1$ and let $d_1$ and $c$ vary. (a-b) When $d_1=0.08, 0.1$ and $ c=1$, species $Y$ goes to extinction and $X$ approaches its carrying capacity. (c) When $d_1=0.2$ and $ c=1$, species $X$ goes to extinction and $Y$ approaches its carrying capacity. (d) When $d_1=0.2$ and $ c=20$, the species coexist

Figure 3.  Phase-plane diagrams of (4) when the mortality $d_1$ varies. Red and blue lines are the isoclines for $x$ and $y$, respectively. Grey arrows display the direction and strength of the vector fields in the phase-plane space. Fix $r_1=r_2=1, d_2=0.1, c=20$. (a) When $d_1(=2)$ is large, species $X$ goes to extinction and $Y$ approaches its carrying capacity. (b) When $d_1(=1.1)$ is intermediate, the species coexist. (c) When $d_1(=0.2)$ is small, the species coexist and species $Y$ approaches a density extremely less than its carrying capacity. (d) When $d_1(=0.08)$ is extremely small, species $Y$ goes to extinction and $X$ approaches its carrying capacity

Figure 4.  Bifurcation diagram of system (4) on the $d_1-c$ plane. Fix $r_1=r_2=1, d_2=0.2$. Then lines $d_1=0.2, d_1=1.0, c=6.25 (d_1-0.2)$ and $c=5$ divide the first quadrant into 6 regions. In the region $d_1\le 0.2$, the interaction outcome remains amensalism $(0~-)$. In the regions $0.2< d_1< 1.0$, the interaction outcome changes from amensalism $(-~0)$, to competition $(-~-)$, to the other amensalism $(-~0)$ and to parasitism $(+~-)$ in a smooth fashion when the facilitation $c$ increases. Similarly, in the regions $d_1\ge 1.0$, the interaction outcome changes from neutralism $(0~0)$ to parasitism $(+~-)$ in a smooth fashion when the facilitation $c$ increases