Phase Multistability of Dynamics Modes of the Ricker Model with Periodic Malthusian Parameter

Abstract

The paper investigates the phase multistability of dynamical modes of the Ricker model with 2-year periodic Malthusian parameter. It is shown that both the variable perturbation and the phase shift of the Malthusian parameter can lead to a phase shift or a change in the dynamic mode observed. The possibility of switches between different dynamic modes is due to multistability, since the model has two different stable 2-cycles. The first stable 2-cycle is the result of transcritical bifurcation and is synchronous to the oscillations of the Malthusian parameter. The second stable 2-cycle arises as a result of the tangent bifurcation and is asynchronous to the oscillations of the Malthusian parameter. This indicates that two-year fluctuations in the population size can be both synchronous and asynchronous to the fluctuations in the environment. The phase shift of the Malthusian parameter causes a phase shift in the stable 4-cycle of the first bifurcation series to one or even three elements of the 4-cycle. The phase shift to two elements of this 4-cycle is possible due to a change in the half-amplitude of the Malthusian parameter oscillation or the variable perturbation. At the same time, the longer period of the cycle, the more phases with their attraction basins it has, and the smaller the threshold values above which shift from the attraction basin to another one occur. As a result, in the case of cycles with long period (for example, 8-cycle) perturbations, that stable cycles with short period are able to "absorb", can cause different phase transitions, which significantly complicates the dynamics of the model trajectory and, as a consequence, the identification of the dynamic mode observed.

Transition from Bi- to Quadro-Stability in Models of Population Dynamics and Evolution

“…In other discrete-time population models, the general asymmetric scenario of the transition to quadro-stability often arises. For example, in the Ricker equation with a periodic population growth rate, there is a transition from the phase bistability of the 2-cycle to the quadro-stability [61,62], which leads to anti-phase 2-cycles. However, this scenario is more similar to the scenario of the formation of quadro-stability in the model of the dynamics of two populations coupled by migration, functioning according to the Bazykin Equation (model ( 9)).…”

Mathematical Modeling of Population Dynamics Based on Recurrent Equations: Results and Prospects. Part I

“…Of special interest are periodic changes in environmental factors, which, in particular, may lead to periodic changes in the reproductive potential of populations with a seasonal reproduction pattern. This situation can be described using the Ricker equation with a periodic Malthusian parameter, which makes it possible to take into account the cyclical effects of both exogenous and endogenous factors on the population size (Shlyufman et al, 2016(Shlyufman et al, , 2017(Shlyufman et al, , 2018Frisman et al, 2019). The study showed that the Ricker equation with a Malthusian parameter varying with a period of two, along with the multistability of dynamic modes, also has phase multistability.…”