Evolution towards oscillation or stability in a predator-prey system

Abstract

We studied a prey-predator system in which both species evolve. We discuss here the conditions that result in coevolution towards a stable equilibrium or towards oscillations. First, we show that a stable equilibrium or population oscillations with small amplitude is likely to occur if the prey's (host's) defence is effective when compared with the predator's (parasite's) attacking ability at equilibrium, whereas large-amplitude oscillations are likely if the predator's (parasite's) attacking ability exceeds the prey's (host's) defensive ability. Second, a stable equilibrium is more likely if the prey's defensive trait evolves faster than the predator's attack trait, whereas population oscillations are likely if the predator's trait evolves faster than that of the prey. Third, when the adaptation rates of both species are similar, the amplitude of the fluctuations in their abundances is small when the adaptation rate is either very slow or very fast, but at an intermediate rate of adaptation the fluctuations have a large amplitude. We also show the case in which the prey's abundance and trait fluctuate greatly, while those of the predator remain almost unchanged. Our results predict that populations and traits in host-parasite systems are more likely than those in prey-predator systems to show large-amplitude oscillations.

Figure 1.

An example of non-equilibrium dynamics in relation to the speed of adaptation. We adopted the linear functions r = r₀(1−ρ_Xu) and g = g₀(1−ρ_Y v), where r₀ and g₀ are the basal per capita prey growth rate and the basal conversion efficiency of the predator, respectively, and ρ_X and ρ_Y represent the strength of the trade-off in the prey and the predator, respectively. We assumed that . The solid and dotted lines indicate the dynamics of the prey and predator, respectively. The inset in the lower panel of (a) enlarges a portion of panel. The parameter values are ρ_X = ρ_Y = 2, r₀ = 1, g₀ = 1, a₀ = 3, θ = 20 and d = 0.3. The initial values are (X, Y, u, v) = (1, 0.5, 0.1, 0.1).

Figure 2.

The parameter regions in which the equilibrium is stable or unstable. The two axes are and . We adopted the linear functions r = r₀(1−ρ_Xu) and g = g₀(1−ρ_Yv). The white and shaded regions indicate the parameter ranges in which the equilibrium is stable and unstable, respectively. (a) u* = v*. We assumed ρ_X = ρ_Y=2. (b) u* < v*. The shaded region is the unstable region when ρ_X = 1.98 and ρ_Y = 2. The ordered pairs show (ρ_X, ρ_Y), corresponding to each boundary (dotted lines) between stable and unstable regions, with the region on the left side of the boundary unstable and the one on the right stable. The other parameter values are r₀ = 1, g₀ = 1, a₀ = 3, θ = 20 and d = 0.2.

Figure 3.

Bifurcation diagrams of (a,b) population and (c) trait dynamics, and (d) the ratio of the amplitudes of oscillation in the two species in relation to the speed of adaptation. The points indicate the minimum and maximum values. The black and grey points in panel (c) are the maximum and minimum trait values in the prey and predator, respectively. The Arabic numerals in square brackets above the panels indicate three phases: [1] trait–abundance cycle; [2] resonance; and [3] trait–trait cycle phases (see text). Other parameter values are ρ_X = ρ_Y = 1, r₀ = 1, g₀ = 1, a₀ = 6, θ = 40 and d = 0.45. The initial values are (X, Y, u, v) = (1, 0.5, 0.1, 0.1).

Figure 4.

Bifurcation diagrams of (a,b) population and (c) trait dynamics, and (d) the ratio of the amplitudes of oscillation in the two species in relation to the speed of adaptation. The Arabic numerals in square brackets above the panels indicate phases, as in figure 3; [4] indicates the stationary phase (see text). Parameters are the same as in figure 2 except for ρ_X = 1 and ρ_Y = 1.01.