Fear factor in a prey–predator system in deterministic and stochastic environment

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

“…where 3 , 𝑓 (2) 02 = −2h, 𝑓 (2) 11 = pc (1 + mX s ) 2 , 𝑓 (2) 30 = 6pcm 2 Y s (1 + mX s ) 4 , 𝑓 (2) 21 = − 2pcm (1 + mX s ) 3 , 𝑓 (2) 12 = 0, 𝑓 (2) 03 = 0.…”

“…$$ Although there are several ways to introduce fear in the model mathematically, here it is considered that $$$ F\left(Y,\upsilon \right)=\frac{1}{1+\upsilon Y} $$$, where $$$ \nu $$$ denotes fear level, provided by Mukherjee 2 . See Das and Samanta, Mukherjee, Roy and Alam, and Sasmal and Takeuchi 1–3,11 for discussions on recent papers with other functional forms of fear factor. Various other parameters are used in the model: $$$ r $$$ (birth rate of prey), $$$ \alpha $$$ (natural death rate for prey), $$$ a $$$ (rate for intra‐species competition), $$$ p $$$ (per capita consumption rate for predator), $$$ b $$$ (fear level associated with the death rate of prey ), $$$ m $$$ (handling time for captured prey species), $$$ c $$$ (conversion efficiency, namely, the ability of predators converting prey biomass into new predators), $$$ d $$$ (natural death rate for predator), $$$ h $$$ (intra‐specific competition that takes place in the predator population), and $$$ \eta $$$ (the level of time delay).…”

Exploring a Simple Stochastic Mathematical Model Including Fear with a Linear Functional Response

“…Hence incorporating random noise in model parameters may significantly alter dynamics of both prey and predator species. Although any parameter of the model may be affected by environmental noise, the uncertain growth and death rates may be particularly influenced, see for example [16,21,22].…”

“…Therefore, it is worth to incorporate environmental noise in the model to better capture the dynamics of species in ecology. Many scientist have studied models with Gaussian noise from various point of view, including stochastic models with stage structure [14], foraging [15], anti-predator defence [13], intra-specific competition [16], leading to stochastic differential equations. This paper is organised as follows: The deterministic model given in [17] is revisited in Section 2, where positivity of the model, its steady states and local stability analysis are derived respectively in Sections 2.1, 2.2 and 2.3.…”

The Long Time Behavior of Equilibrium Status of a Predator‐Prey System with Delayed Fear in Deterministic and Stochastic Scenarios

“…In real world, when prey populations perceive some predation risks, they will always show a few anti-predator actions, such as vigilance, foraging actions, and some psychological changes, which cut down the growth speed of the prey species and the survival speed of adults is also impacted accordingly. Recently, many predator-prey population systems containing fear of prey species are studied and some valuable results are reported [7][8][9].…”