Ecology and evolution of symbiosis in metapopulations

Figures & data

Table 1. Interior equilibria and bifurcation by p.

Case 1
	There is no interior equilibrium
Case 2
	There is one interior equilibrium
Case 3	p∈[0, p 0)			p∈[p 0, 1]
	No equilibrium			One equilibrium
Case 4	p∈[0, p 0)	p = p 0	p∈(p 0, 1]
	No equilibrium	One equilibrium	Two equilibria
Case 5	p∈[0, p 1)	p = p 1	p∈(p 1, p 2)	p∈[p 2, 1]
	No equilibrium	One equilibrium	Two equilibria	One equilibrium

Figure 1. No interior equilibria. The symbiont-free equilibrium is stable and uninvadable for all values of p∈[0, 1].

Figure 2. The symbiont-free equilibrium is unstable. The unique coexistence equilibrium is stable for all values of p∈[0, 1]. Arrows indicate evolution of p as strategy of the plant or symbiont. The plant and the symbiont have opposite interest, their evolution pulls p in opposite senses.

Figure 3. The symbiont-free equilibrium stable on [0, 0.4584) bifurcates to the unique stable coexistence equilibrium. Arrows indicate evolution of p as strategy of the plant or symbiont.

Figure 4. The symbiont-free equilibrium stable on [0,1]. Two interior equilibria on [0.9769,1], one stable, one unstable. Arrows indicate evolution of p as strategy of the plant or symbiont. Evolutionary conflict between plant and symbiont on [0.9769,0.99553], agreement on [0.99553,1].

Figure 5. The symbiont-free equilibrium stable on (0, 0.787). One unstable interior equilibrium on (0.7659,0.787), one stable interior equilibrium on (0.7659, 1). Arrows indicate evolution of p as strategy of the plant or symbiont.