A type IV functional response with different shapes in a predator–prey model

Köhnke, Merlin C.; Siekmann, Ivo; Seno, Hiromi; Malchow, Horst

doi:10.1016/j.jtbi.2020.110419

Cited by 21 publications

(18 citation statements)

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“…where is expressed in Eq. (41). Then x((n + k)T) > x(nT) k → ∞ as k → ∞, which contradicts the boundedness of x(t).…”

Section: Extinction and Permanencementioning

confidence: 93%

“…Some interesting functional responses, widely used in predator-prey models, are Holling type, sigmoid type and Beddington-DeAngelis functional responses. More details as regards the functional responses can be found in [23,[39][40][41]. However, the generalized Holling type IV functional response or the Monod-Haldane function [5,[42][43][44][45]…”

Section: Introductionmentioning

confidence: 99%

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Investigation on dynamics of an impulsive predator–prey system with generalized Holling type IV functional response and anti-predator behavior

2021

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In the present article, we propose and analyze a new mathematical model for a predator–prey system including the following terms: a Monod–Haldane functional response (a generalized Holling type IV), a term describing the anti-predator behavior of prey populations and one for an impulsive control strategy. In particular, we establish the existence condition under which the system has a locally asymptotically stable prey-eradication periodic solution. Violating such a condition, the system turns out to be permanent. Employing bifurcation theory, some conditions, under which the existence and stability of a positive periodic solution of the system occur but its prey-eradication periodic solution becomes unstable, are provided. Furthermore, numerical simulations for the proposed model are given to confirm the obtained theoretical results.

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“…where is expressed in Eq. (41). Then x((n + k)T) > x(nT) k → ∞ as k → ∞, which contradicts the boundedness of x(t).…”

Section: Extinction and Permanencementioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Investigation on dynamics of an impulsive predator–prey system with generalized Holling type IV functional response and anti-predator behavior

2021

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“…As we consider a predator-prey model with group defense in the prey, we use the kinetic equations developed in Köhnke et al (2020)…”

Section: Model and Methodsmentioning

confidence: 99%

Taxis-driven pattern formation in a predator-prey model with group defense

Köhnke

Siekmann

Malchow

2020

Ecological Complexity

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We consider a reaction-diffusion(-taxis) predator-prey system with group defense in the prey. Taxis-driven instability can occur if the group defense influences the taxis rate (Wang et al., 2017). We elaborate that this mechanism is indeed possible but biologically unlikely to be responsible for pattern formation in such a system. Conversely, we show that patterns in excitable media such as spatiotemporal Sierpinski gasket patterns occur in the reactiondiffusion model as well as in the reaction-diffusion-taxis model. If group defense leads to a dome-shaped functional response, these patterns can have a rescue effect on the predator population in an invasion scenario. Preytaxis with prey repulsion at high prey densities can intensify this mechanism leading to taxis-induced persistence. In particular, taxis can increase parameter regimes of successful invasions and decrease minimum introduction areas necessary for a successful invasion. Last, we consider the mean period of the irregular oscillations. As a result of the underlying mechanism of the patterns, this period is two orders of magnitude smaller than the period in the nonspatial system. Counter-intuitively, faster-moving predators lead to lower oscillation periods and eventually to extinction of the predator population. The study does not only provide valuable insights on theoretical spatially explicit predator-prey models with group defense but also comparisons of ecological data with model simulations.

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“…Recently, a series of mathematical models has been developed to better understand the dynamics of the predator-prey system with different kinds of functional responses [6,[16][17][18][19][20][21][22][23][24]. Dalziel et al [16] studied a modified Holling type-II predator-prey model, based on the premise that the search rate of predators is dependent on the prey density, rather than constant.…”

Section: Introductionmentioning

confidence: 99%

“…ey also defined the basic reproduction number R 0 and show that it is a threshold parameter determining whether the predators can coexist with the prey species. Kohnke et al [18] studied a predatorprey model and developed a functional response using timescale separation. In their study, they showed that the resulting functional response contains a single parameter that controls whether the group defense functional response is saturating or dome-shaped.…”

Section: Introductionmentioning

confidence: 99%

Rich Dynamics of a Predator-Prey System with Different Kinds of Functional Responses

et al. 2020

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In this study, we investigate a mathematical model that describes the interactive dynamics of a predator-prey system with different kinds of response function. The positivity, boundedness, and uniform persistence of the system are established. We investigate the biologically feasible singular points and their stability analysis. We perform a comparative study by considering different kinds of functional responses, which suggest that the dynamical behavior of the system remains unaltered, but the position of the bifurcation points altered. Our model system undergoes Hopf bifurcation with respect to the growth rate of the prey population, which indicates that a periodic solution occurs around a fixed point. Also, we observed that our predator-prey system experiences transcritical bifurcation for the prey population growth rate. By using normal form theory and center manifold theorem, we investigate the direction and stability of Hopf bifurcation. The biological implications of the analytical and numerical findings are also discussed in this study.

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A type IV functional response with different shapes in a predator–prey model

Abstract: The version presented here may differ from the published version or from the version of the record. Please see the repository URL above for details on accessing the published version and note that access may require a subscription.

Cited by 21 publications

References 50 publications

Investigation on dynamics of an impulsive predator–prey system with generalized Holling type IV functional response and anti-predator behavior

Investigation on dynamics of an impulsive predator–prey system with generalized Holling type IV functional response and anti-predator behavior

Taxis-driven pattern formation in a predator-prey model with group defense

Rich Dynamics of a Predator-Prey System with Different Kinds of Functional Responses

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