Population cycles emerging through multiple interaction types

Abstract

Cyclic dynamics of populations are outstanding and widespread phenomena across many taxa. Previous theoretical studies have mainly focused on the consumer-resource interaction as the driving force for such cycling. However, natural ecosystems comprise diverse types of species interactions, but their roles in population dynamics remains unclear. Here, using a four-species hybrid module with antagonistic, mutualistic and competitive interactions, we analytically showed that the system with major interaction types can drive population cycles. Stronger interactions easily cause cycling, and even when sub-modules with possible combinations of two interactions are stabilized by weak interactions, the system with all interaction types can cause unstable population oscillations. Diversity of interaction types allows to add mutualists to the list of drivers of oscillations in a focal species' population size, when they act in conjunction to other drivers.

Keywords: competition; cycling; mathematical model; mutualism; predator–prey; stability.

Figure 1.

Dynamics of population sizes. Illustrations along the upper side of the panels indicate the community modules that cause each population dynamics. The signs within the linking lines indicate the antagonistic (±), mutualistic (+) and competitive (–) interaction types. Colours indicate different species. Parameter values are: r_X = r_Z = r_W = 1, g = 0.25, d = 0.05, h_X= h_Z= 1, a = 1.8, u = 3, v = 2, α = 0.2 and β = 0.7. We chose this to satisfy the condition under which the sub-modules are under stable equilibrium, but the mixture of such modules causes instability (in equations (3.1)–(3.4)). Even if the strong symmetrical parameter values are relaxed, the same hybrid effect can be found if these inequalities are balanced.

Figure 2.

Effects of interspecific competition on the amplitudes and coefficient of variation of population oscillations. The amplitude (solid line) and coefficient of variation (dashed line) of population oscillation were calculated after using the dynamic approach to asymptotic behaviours, with varying competition coefficients. The resource species is used as the index of the amplitude. The tendency was the same regardless of the species used. Parameter values are the same as in figure 1 except for a = 2.5.

Figure 3.

Parameter regions indicating stability of the coexistence equilibrium. Coexistence is impossible in the black region. However, the coexistence equilibrium is stable in the white region and unstable (a limit cycle occurs) in the grey region. The values of competition coefficients (non-free parameters) are shown above the panels. Parameter values are the same as in figure 1.