Dynamics and spatio-temporal patterns in a prey–predator system with aposematic prey

Sourav Kumar Sasmal; Jeet Banerjee; Yasuhiro Takeuchi; Sourav Kumar Sasmal; Jeet Banerjee; Yasuhiro Takeuchi

doi:10.3934/mbe.2019191

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 3864-3884. doi: 10.3934/mbe.2019191

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Dynamics and spatio-temporal patterns in a prey–predator system with aposematic prey

1.
Department of Physics and Mathematics, Aoyama Gakuin University, Kanagawa 252-5258, Japan
2.
Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India

Received: 20 February 2019 Accepted: 26 April 2019 Published: 01 May 2019

We analyze the impact of aposematic time and searching efficiency of prey on the temporal and spatio-temporal dynamics of a diffusive prey-predator system. Here, our assumption is that the prey population primarily invests its total time in two activities——(ⅰ) defense against predation and (ⅱ) searching for food, followed by growth-induced reproduction, whereas, predators do not involve in self-defense. Moreover, we consider that the reproduction rate of prey and the rate of predation have a negative linear correlation with the amount of time invested for aposematism. Based on the presumptions, we find that unlike searching efficiency of prey, the aposematic time can diminish the proportion in which prey and predator coexist when it crosses a certain threshold, and at the extreme aposematism, the entire population drives into the extinction. The proposed dynamics undergoes Hopf-bifurcation with respect to the searching efficiency of prey. We examine the individual effect of aposematic time and searching efficiency on the formation of regular Turing patterns——the low to medium to high values of defense-time and food searching efficiency generate 'spots' to 'stripes' to 'holes' pattern, respectively; however, the combined impact of both presents only non-Turing 'spot' pattern with the 'predominance of predators, ' which happens through the Turing-Hopf bifurcation.
- aposematism,
- searching efficiency,
- turing patterns,
- Hopf-bifurcation,
- Turing-Hopf bifurcation,
- Prey–predator system
Citation: Sourav Kumar Sasmal, Jeet Banerjee, Yasuhiro Takeuchi. Dynamics and spatio-temporal patterns in a prey–predator system with aposematic prey[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 3864-3884. doi: 10.3934/mbe.2019191

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Abstract

We analyze the impact of aposematic time and searching efficiency of prey on the temporal and spatio-temporal dynamics of a diffusive prey-predator system. Here, our assumption is that the prey population primarily invests its total time in two activities——(ⅰ) defense against predation and (ⅱ) searching for food, followed by growth-induced reproduction, whereas, predators do not involve in self-defense. Moreover, we consider that the reproduction rate of prey and the rate of predation have a negative linear correlation with the amount of time invested for aposematism. Based on the presumptions, we find that unlike searching efficiency of prey, the aposematic time can diminish the proportion in which prey and predator coexist when it crosses a certain threshold, and at the extreme aposematism, the entire population drives into the extinction. The proposed dynamics undergoes Hopf-bifurcation with respect to the searching efficiency of prey. We examine the individual effect of aposematic time and searching efficiency on the formation of regular Turing patterns——the low to medium to high values of defense-time and food searching efficiency generate 'spots' to 'stripes' to 'holes' pattern, respectively; however, the combined impact of both presents only non-Turing 'spot' pattern with the 'predominance of predators, ' which happens through the Turing-Hopf bifurcation.

References

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