Last modified: 2014-03-28

#### Abstract

In this paper we consider evolutionary scenarios of the origination of oscillatory and chaotic population dynamics in the species with the simple age structure. We take into account nonlinear interactions, observed in nature, between different age groups in the population, influencing the population’s birth rate. We consider the population which, by the end of each reproduction season, consists of two age groups: juveniles (immature individuals) and adults (participants of the reproduction process). We assume that the time between two reproduction seasons is enough for the juveniles to become adults.

It is made detailed analytical investigation of the model and its simulation. It is shown this mathematical system has unique nontrivial solution. Depending on the values of model parameters we have found its stability conditions. The stability of the non-trivial solution is defined by its eigenvalues, which are solutions of the characteristic equation for the model. We investigate the dynamic modes of the model (scenarios of transition from equilibrium to the irregular dynamics). A series of simulations showed that several attractors with their basins of attraction can coexist in the phase space of the model. In particular, we found a sub-area into the stability domain of the unique non-trivial stationary solution where a new stable attractor appeared. This sub-area has a cycle length of three. Both attractors were also found in this sub-area. A cycle length of three is the result of tangent bifurcation. On the basis of constructed bifurcation diagrams, attraction basins and maps of asymptotic dynamic modes for the model it is made influence research of initial conditions on the system behavior.

In addition, we have made a detailed analysis of the effect of intraspecific competition between the population age groups on occurrence of number fluctuations. Our analysis shows that birth rate’s decrease with the number of adults is an efficient mechanism for controlling the population size. Through the growth of the individual reproductive potential it can lead to oscillations of population size with fairly complex temporal structure. If the birth rate also is controlled by the number of juveniles then such a mechanism positively affects stability only if the dependence on the number of juveniles is modest and is weaker than the dependence on the number of adults. If these requirements are met, the stability domain increases substantially. The regulation of the birth rate by the number of juveniles appears to be inefficient; small increase of the reproductive potential allows the population to start growing exponentially, which leads to formation of new restricting mechanisms.

Thus the proposed model has several stable attractors, particularly cycle lengths of three arising from tangential bifurcation. Therefore, the population dynamics significantly depended on the initial conditions (or the current values of numbers). It shown the influence of external factors on population reproduction significantly expands the range of possible dynamic modes, and leads a random walk into the attraction basins of their modes.