Abstract
In this study, the different dynamical behaviors caused by different parameters of a discrete-time eco-epidemiological model with disease in prey are discussed in ecological perspective. The results indicate that when we choose the same parameters and initial value and only vary the key parameters there appears a series of dynamical behaviors. For example, only varying the death rate of the infected prey (the carrying capacity of the environment for the prey population or the transmission coefficient), there appear chaos, Hopf (flip) bifurcation, local stability, flip (Hopf) bifurcation, and chaos; when only varying the predation coefficient there appear chaos, Hopf bifurcation, local stability, Hopf bifurcation, and chaos. These results are far richer than the corresponding continuous-time model and are rarely seen in previous works. Numerical simulations not only illustrate our results but also exhibit complex dynamical behaviors, such as period-doubling bifurcation in period-2,4,8, quasi-periodic orbits, 3,5,11,16-period orbits and chaotic sets. Moreover, the numerical simulations imply that when the death rate of the infected prey reaches a fixed value the disease dies out. Also, when the predation coefficient parameter reaches some value the disease dies out. These findings indicate that it is practicable to control the disease transmitting in prey by changing the death rate of the infected prey and the predation coefficient parameter.
Highlights
The ecological models are used to study the competitive, cooperation, and preypredator relationships between different species in nature [ – ]
In Section we present the numerical simulations, which illustrate our results with the theoretical analysis, and exhibit the complex dynamical behaviors such as the invariant cycle, -periodic solutions, flip bifurcation, Hopf bifurcation, and more than one attractors and chaotic sets
From Figure, we find that there exists a Hopf bifurcation and chaos of model ( . )
Summary
The ecological models are used to study the competitive, cooperation, and preypredator relationships between different species in nature [ – ]. We can control the transmission of diseases among different species through varying the key parameters when the dynamical behaviors of the corresponding models have been discussed clearly [ , ]. The dynamical behaviors of the models with disease are studied, such as the stability, periodic solution, oscillation, bifurcation, and chaos. Their results indicated that the predators die out and the prey tends to its carrying capacity; or the infected prey and the predators both die out; or the predator and prey coexist. The complex dynamical behaviors of discrete-time predator-prey models have already received much attention by lots of studies: such as stability, permanence, existence of periodic solutions, bifurcation, and chaos phenomenons [ – ].
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