Diffusion-induced regular and chaotic patterns in a ratio-dependent predator–prey model with fear factor and prey refuge

Abstract

Population distribution of interacting species in a large scale natural system is heterogeneous and subject to change for various reasons. Here, we explore how behavioral modification in prey species due to fear of predator and mutual interference between predators can create different spatiotemporal patterns in population distribution. We show that the fear factor and diffusion in a ratio-dependent predator–prey model may show more complex dynamics than observed earlier. It is shown that when prey diffusivity is low, prey remains concentrated at different patches throughout the domain. However, prey density becomes low at the patches as they disperse at a higher rate. Mixed and stripe patterns are observed during the transition from a hot spot pattern at the lower prey diffusivity to a cold spot pattern at its higher value. Pattern transition is, however, completely opposite if the antipredator behavior is gradually increased. Our simulation results reveal that the spatiotemporal chaotic pattern may also be observed in the Hopf–Turing region of instability provided prey shows a higher level of antipredator behavior. The chaotic pattern of the Hopf–Turing region may be shifted to a spot type pattern of the Turing region depending on the refuge level of the habitat.

Pattern formation of parasite-host model induced by fear effect

“…Through the observation of the pattern growth, we find some interesting phenomena. The results show that under the effect of fear, the pattern growth mechanism is always unified, growing in the form of cold spots → cold spots-stripes → cold stripes → hot stripes → hot spots-stripes → hot spots, which is also consistent with the results of literature 17 . Moreover, we also find that the change of pattern growth caused by natural mortality was opposite to that of fear factor.…”

“…However, researches on the change of spatial population dynamics caused by fear effect are still not general. Through the description of 17,27 , we can find that the evolution of pattern induced by fear effect often follows the rule that with the increase of fear, the growth order of pattern is: cold spots → cold spots-stripes → cold stripes → hot stripes → hot spots-stripes → hot spots. As for whether such a conclusion is general, we need to continue to explore in our future work.…”

Pattern formation of parasite-host model induced by fear effect

“…Through the observation of the pattern growth, we find some interesting phenomena. The results show that under the effect of fear, the pattern growth mechanism is always unified, growing in the form of cold spots → cold spots-stripes → cold stripes → hot stripes → hot spots-stripes → hot spots, which is also consistent with the results of literature 17 . Moreover, we also find that the change of pattern growth caused by natural mortality was opposite to that of fear factor.…”

“…However, researches on the change of spatial population dynamics caused by fear effect are still not general. Through the description of 17,27 , we can find that the evolution of pattern induced by fear effect often follows the rule that with the increase of fear, the growth order of pattern is: cold spots → cold spots-stripes → cold stripes → hot stripes → hot spots-stripes → hot spots. As for whether such a conclusion is general, we need to continue to explore in our future work.…”

Turing patterns by supramolecular self-assembly of a single salphen building block

“…These principles have been observed not only in biological morphogenetic processes but are also widely present in many more dynamic systems than thought earlier ( Kuznetsov and Polezhaev, 2020 ; Leyshon et al., 2021 ; Scholes et al., 2019 ), including demographic, sociolinguistic, psychologic, economic, ecologic, or epidemiologic phenomena ( Batabyal, 2021 ; Chakraborty et al., 2021 ; Chen, 2019 ; Iskarous, 2019 ; Lacalli, 2020 ; Mimar et al., 2021 ; Pal and Poria, 2021 ; Putra et al., 2019 ; Vandermeer and Perfecto, 2020 ; Zincenko et al., 2021 ). Nevertheless, in chemistry, these RD principles have been typically used to explain how reactions concerted at different scales can result into oscillatory and out-of-equilibrium systems, such as the Belousov-Zhabotinsky (BZ) reaction ( Zaikin and Zhabotinsky, 1970 ) ( e .…”

Complex dynamics of a predator-prey model with hunting cooperation-induced fear and diffusion

“…In addition, Sarkara and Khajanchi [23] proposed a predator-prey system introducing the cost of fear into prey reproduction with Holling type-II functional response, which found that the sensitivity of prey to perceive the risk of predation increases with the growth rate of prey or the decay rate of predator. And there are other papers that also consider the influence on the fear effect on the predator-prey models [24][25][26][27][28][29][30][31][32][33]. The above analysis shows that the fear effect and hunting cooperation have a fundamental impact on the dynamics of the predator system.…”

Pattern formation of parasite-host model induced by fear effect

“…Through the observation of the pattern growth, we find some interesting phenomena. The results show that under the effect of fear, the pattern growth mechanism is always unified, growing in the form of cold spots → cold spots-stripes → cold stripes → hot stripes → hot spots-stripes → hot spots, which is also consistent with the results of literature 17 . Moreover, we also find that the change of pattern growth caused by natural mortality was opposite to that of fear factor.…”

“…However, researches on the change of spatial population dynamics caused by fear effect are still not general. Through the description of 17,27 , we can find that the evolution of pattern induced by fear effect often follows the rule that with the increase of fear, the growth order of pattern is: cold spots → cold spots-stripes → cold stripes → hot stripes → hot spots-stripes → hot spots. As for whether such a conclusion is general, we need to continue to explore in our future work.…”

Turing patterns by supramolecular self-assembly of a single salphen building block

“…These principles have been observed not only in biological morphogenetic processes but are also widely present in many more dynamic systems than thought earlier ( Kuznetsov and Polezhaev, 2020 ; Leyshon et al., 2021 ; Scholes et al., 2019 ), including demographic, sociolinguistic, psychologic, economic, ecologic, or epidemiologic phenomena ( Batabyal, 2021 ; Chakraborty et al., 2021 ; Chen, 2019 ; Iskarous, 2019 ; Lacalli, 2020 ; Mimar et al., 2021 ; Pal and Poria, 2021 ; Putra et al., 2019 ; Vandermeer and Perfecto, 2020 ; Zincenko et al., 2021 ). Nevertheless, in chemistry, these RD principles have been typically used to explain how reactions concerted at different scales can result into oscillatory and out-of-equilibrium systems, such as the Belousov-Zhabotinsky (BZ) reaction ( Zaikin and Zhabotinsky, 1970 ) ( e .…”

Complex dynamics of a predator-prey model with hunting cooperation-induced fear and diffusion

“…In addition, Sarkara and Khajanchi [23] proposed a predator-prey system introducing the cost of fear into prey reproduction with Holling type-II functional response, which found that the sensitivity of prey to perceive the risk of predation increases with the growth rate of prey or the decay rate of predator. And there are other papers that also consider the influence on the fear effect on the predator-prey models [24][25][26][27][28][29][30][31][32][33]. The above analysis shows that the fear effect and hunting cooperation have a fundamental impact on the dynamics of the predator system.…”

Pattern formation of parasite-host model induced by fear effect

“…Through the observation of the pattern growth, we find some interesting phenomena. The results show that under the effect of fear, the pattern growth mechanism is always unified, growing in the form of cold spots → cold spots-stripes → cold stripes → hot stripes → hot spots-stripes → hot spots, which is also consistent with the results of literature 17 . Moreover, we also find that the change of pattern growth caused by natural mortality was opposite to that of fear factor.…”

“…However, researches on the change of spatial population dynamics caused by fear effect are still not general. Through the description of 17,27 , we can find that the evolution of pattern induced by fear effect often follows the rule that with the increase of fear, the growth order of pattern is: cold spots → cold spots-stripes → cold stripes → hot stripes → hot spots-stripes → hot spots. As for whether such a conclusion is general, we need to continue to explore in our future work.…”

Turing patterns by supramolecular self-assembly of a single salphen building block

“…These principles have been observed not only in biological morphogenetic processes but are also widely present in many more dynamic systems than thought earlier ( Kuznetsov and Polezhaev, 2020 ; Leyshon et al., 2021 ; Scholes et al., 2019 ), including demographic, sociolinguistic, psychologic, economic, ecologic, or epidemiologic phenomena ( Batabyal, 2021 ; Chakraborty et al., 2021 ; Chen, 2019 ; Iskarous, 2019 ; Lacalli, 2020 ; Mimar et al., 2021 ; Pal and Poria, 2021 ; Putra et al., 2019 ; Vandermeer and Perfecto, 2020 ; Zincenko et al., 2021 ). Nevertheless, in chemistry, these RD principles have been typically used to explain how reactions concerted at different scales can result into oscillatory and out-of-equilibrium systems, such as the Belousov-Zhabotinsky (BZ) reaction ( Zaikin and Zhabotinsky, 1970 ) ( e .…”

Complex dynamics of a predator-prey model with hunting cooperation-induced fear and diffusion

“…In addition, Sarkara and Khajanchi [23] proposed a predator-prey system introducing the cost of fear into prey reproduction with Holling type-II functional response, which found that the sensitivity of prey to perceive the risk of predation increases with the growth rate of prey or the decay rate of predator. And there are other papers that also consider the influence on the fear effect on the predator-prey models [24][25][26][27][28][29][30][31][32][33]. The above analysis shows that the fear effect and hunting cooperation have a fundamental impact on the dynamics of the predator system.…”