Cross-Diffusion-Driven Instability in a Predator-Prey System with Fear and Group Defense

Carfora, Maria Francesca; Torcicollo, Isabella

doi:10.3390/math8081244

Cited by 12 publications

(4 citation statements)

References 35 publications

(55 reference statements)

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“…In some cases, the explicit introduction of a space variable can modify the forecasts of the non-spatial counterpart model. The spatial diffusion plays a relevant role in the process of population evolution and in many other fields of applied mathematics ( [35][36][37][38][39][40][41][42][43][44][45]); spatial patterns are of high relevance and importance as they may help to better describe the spatial and temporal distribution of the involved populations, thus providing important hints on the behavior of communities. Motivated by such considerations, we take into account the following system:…”

Section: Introductionmentioning

confidence: 99%

Turing Instability and Spatial Pattern Formation in a Model of Urban Crime

Torcicollo,

Vitiello

2024

Mathematics

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A nonlinear crime model is generalized by introducing self- and cross-diffusion terms. The effect of diffusion on the stability of non-negative constant steady states is applied. In particular, the cross-diffusion-driven instability, called Turing instability, is analyzed by linear stability analysis, and several Turing patterns driven by the cross-diffusion are studied through numerical investigations. When the Turing–Hopf conditions are satisfied, the type of instability highlighted in the ODE model persists in the PDE system, still showing an oscillatory behavior.

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Section: Introductionmentioning

confidence: 99%

Turing Instability and Spatial Pattern Formation in a Model of Urban Crime

Torcicollo,

Vitiello

2024

Mathematics

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“…These factors are mostly related to behavioral pattern of prey species including fear, habitat change, and foraging 2,4,5 . Therefore, decrease in the density of prey species may not only arise from direct predation but also may result from physiological changes leading to anti‐predator response in prey species 6–8 . The role of fear is particularly important since the amount of time spent by prey species for being vigilant with fear is more than that for foraging or searching new habitats 7 .…”

Section: Introductionmentioning

confidence: 99%

A mathematical model of population dynamics revisited with fear factor, maturation delay, and spatial coefficients

Gökçe

2022

Math Methods in App Sciences

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This study concentrates on the dynamics of a prey-predator model incorporating the fear effect in the birth and death rate of prey, whose physiological changes may give rise to undirect predation. In the presence and absence of time delay, single parameter numerical continuation with respect to two parameters, that are (i) fear level in the death rate of prey and (ii) conversion efficiency by which new predators are introduced through prey consumption in the system, is performed. Basic results on extinction and delay-driven Hopf bifurcation criteria are investigated. Then, the model is extended to involve the spatial dynamics with and without time delay. Theoretical results for orientation and stability of Hopf bifurcation in spatial system are provided by applying the normal form recipe and also the center manifold theory. Classical reaction-diffusion-type models, incorporating self-diffusion, can induce regular (periodic) and irregular (chaotic) oscillations in space. However, space periodic oscillations are not common in prey-predator interactions. Here, it is shown that the dynamics of only diffusion involved model is periodically arranged in space and time. However, introducing a very small value of time delay in predator maturation, spatial dynamics with chaos is initiated as a result of the joint effect of time delay and diffusion. This reassures that time delay has a crucial role in population dynamics incorporated with the role of indirect predation and gives some useful intuition into underlying species interactions. Theoretical results of the model in this paper are supported with numerical experiments.

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“…The authors considered, in previous studies, some interacting population models that explored the mentioned characteristics (limited resources, nonlinear growth, fear effect, cooperative behavior), also in the presence of spatial dishomogeneity [13][14][15][16][17]. In the above-mentioned research, systems of both ordinary and partial differential equations have been studied.…”

Section: Introductionmentioning

confidence: 99%

“…Precisely, ODE descriptions for the dynamics of an intraguild predator-prey model [13] and for the spreading of waterborne diseases [15,17] have been considered and the stability of the model solutions discussed. Furthermore, in order to highlight how the spatial diffusion can both play an important role in the population evolution and lead to the formation of spatial patterns, a reaction-diffusion system modeling hunting cooperation [14] and a predator-prey system with fear and group defense [16] have been analyzed. In such researches, the effect of spatial diffusion on the stability of the equilibria has been highlighted and the conditions on the parameters ensuring the formation of patterns (stripes, spots, mixed, etc.)…”

Section: Introductionmentioning

confidence: 99%

A Fractional-in-Time Prey–Predator Model with Hunting Cooperation: Qualitative Analysis, Stability and Numerical Approximations

Carfora

Torcicollo

2021

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A prey–predator system with logistic growth of prey and hunting cooperation of predators is studied. The introduction of fractional time derivatives and the related persistent memory strongly characterize the model behavior, as many dynamical systems in the applied sciences are well described by such fractional-order models. Mathematical analysis and numerical simulations are performed to highlight the characteristics of the proposed model. The existence, uniqueness and boundedness of solutions is proved; the stability of the coexistence equilibrium and the occurrence of Hopf bifurcation is investigated. Some numerical approximations of the solution are finally considered; the obtained trajectories confirm the theoretical findings. It is observed that the fractional-order derivative has a stabilizing effect and can be useful to control the coexistence between species.

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Cross-Diffusion-Driven Instability in a Predator-Prey System with Fear and Group Defense

Cited by 12 publications

References 35 publications

Turing Instability and Spatial Pattern Formation in a Model of Urban Crime

Turing Instability and Spatial Pattern Formation in a Model of Urban Crime

A mathematical model of population dynamics revisited with fear factor, maturation delay, and spatial coefficients

A Fractional-in-Time Prey–Predator Model with Hunting Cooperation: Qualitative Analysis, Stability and Numerical Approximations

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