Abstract
The formation of self-organized patterns in predator-prey models has been a very hot topic recently. The dynamics of these models, bifurcations and pattern formations are so complex that studies are urgently needed. In this research, we transformed a continuous predator-prey model with Lesie-Gower functional response into a discrete model. Fixed points and stability analyses were studied. Around the stable fixed point, bifurcation analyses including: flip, Neimark-Sacker and Turing bifurcation were done and bifurcation conditions were obtained. Based on these bifurcation conditions, parameters values were selected to carry out numerical simulations on pattern formation. The simulation results showed that Neimark-Sacker bifurcation induced spots, spirals and transitional patterns from spots to spirals. Turing bifurcation induced labyrinth patterns and spirals coupled with mosaic patterns, while flip bifurcation induced many irregular complex patterns. Compared with former studies on continuous predator-prey model with Lesie-Gower functional response, our research on the discrete model demonstrated more complex dynamics and varieties of self-organized patterns.
Highlights
Predator-prey systems are some of the essential ecological systems in Nature
The formation of patterns has become a very hot topic [6,7,8,9]. This is because the formation process demonstrates self-organization of spatial heterogeneity, and shows system complexity directly and visibly. This visible complexity matches well what has been found in real ecosystems, the dynamics of predator-prey systems are so complex that more studies are still needed to explore the mechanism of pattern formation
Based on the approach in prior studies [26,37,38,39], here we investigated the complex dynamics via transforming the continuous model (3) to a discrete model
Summary
Predator-prey systems are some of the essential ecological systems in Nature. The dynamic behaviors of the predator-prey system have captured the interest of both biologists and ecologists [1,2,3,4,5]. This is because the formation process demonstrates self-organization of spatial heterogeneity, and shows system complexity directly and visibly This visible complexity matches well what has been found in real ecosystems, the dynamics of predator-prey systems are so complex that more studies are still needed to explore the mechanism of pattern formation. We will transform a well-recognized continuous predator-prey model into a discrete model. Based on the approach in prior studies [26,37,38,39], here we investigated the complex dynamics via transforming the continuous model (3) to a discrete model. Numerical simulations were carried out under these bifurcation conditions, to show the complex dynamics and the formation self-organized patterns with this discrete model. Discussions focused on the types of self-organized patterns, and the relations between bifurcations and pattern types
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