Narrow Results

In many models of pest control, increases in pest population due to birth are assumed to be continuous, but in fact, pest population reproduces only during a single period; at the same time, pesticides are often applied during the period. So in this paper we propose a ratio-dependent predator–prey model with birth pulse and pesticide pulse. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact periodic solution of systems which have Ricker functions or Beverton–Holt functions, and obtain the threshold conditions for their stability. Above the threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behaviors of the ratio-dependent predator–prey model with birth pulse and pesticide pulse are very complex, including small-amplitude oscillations, large-amplitude cycles and chaos. This suggests that birth pulse and pesticide pulse, in effect, provide a natural period or cyclicity that allows for period-doubling bifurcation and period-halving bifurcation route to chaos.

In this paper, we propose an exploited single-species discrete population model with stage structure for the dynamics in a fish population for which births occur in a single pulse once per time period. Using the stroboscopic map, we obtain an exact cycle of the system, and obtain the threshold conditions for its stability. Bifurcation diagrams are constructed with the birth rate (or harvesting effort) as the bifurcation parameter, and these are observed to display complex dynamic behaviors, including chaotic bands with period windows, pitchfork and tangent bifurcation, nonunique dynamics (meaning that several attractors or attractor and chaos coexist), basins of attraction and attractor crisis. This suggests that birth pulse provides a natural period or cyclicity that makes the dynamical behaviors more complex. Moreover, we show that the timing of harvesting has a strong impact on the persistence of the fish population, on the volume of mature fish stock and on the maximum annual-sustainable yield. An interesting result is obtained that, after the birth pulses, the population can sustain much higher harvesting effort if the mature fish is removed as early in the season as possible.

In this paper, we propose a model for the dynamics of a fatal infectious disease in a wild animal population with birth pulses and pulse culling, where periodic birth pulses and pulse culling occur at different fixed times. Using the discrete dynamical system determined by stroboscopic map, we obtain an exact cycle of the system, and obtain the threshold conditions for its stability. Bifurcation diagrams are constructed with the birth rate (or culling effort) as the bifurcation parameter, and these are observed to display complex dynamic behaviors, including chaotic bands with period windows, period-doubling and period-halving bifurcations, pitch-fork and tangent bifurcations, nonunique dynamics (meaning that several attractors or attractor and chaos coexist), basins of attraction and attractor crisis. This suggests that birth pulses and pulse culling provide a natural period or cyclicity that makes the dynamical behaviors more complex. Moreover, we investigate the sufficient conditions for global stability of semi-trivial periodic solutions.

In most models of population dynamics, increases in population due to birth are assumed to be time dependent, but many species reproduce only a single period of the year. In this paper, we construct a stage-structured pest model with birth pulse and periodic spraying pesticide at fixed time in each birth period by using impulsive differential equation. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact periodic solution of systems which are with Ricker function or Beverton-Holt function, and obtain the threshold conditions for their stability. Further, we show that the time of spraying pesticide has a strong impact on the number of the mature pest population. Our results imply that the best time of spraying pesticide is at the end of the season, that is before and near the time of birth. Finally, by numerical simulations we find that the dynamical behaviors of the stage-structured population models with birth pulse and impulsive spraying pesticide are very complex, including period-doubling cascade, period-halving cascade, chaotic bands with periodic windows and "period-adding" phenomena.

We investigate the dynamics of a single-species model with birth pulses and pulse harvesting in a polluted environment. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact 1-period solution of the system whose birth function is a Ricker or Beverton–Holt function and obtain the threshold conditions for their stability. Further, we show the effects of the time of pulse harvesting on the maximum annual-sustainable yield. Our results show that the best time for harvesting is immediately after the birth pulses. Numerical simulation results also show that birth pulses and pulse harvesting make the single-species model in a polluted environment we consider more complex, and the system is dominated by periodic and chaotic solutions.

In this paper, a stage-structured predator–prey system with birth pulse and disturbed time delay is investigated. The conditions of the prey-extinction periodic solution of the system which are globally attractive have been obtained. Furthermore, the sufficient conditions for the permanence of the system are established. Finally, numerical analysis is given to confirm the theoretical results.