Why chaos is rarely observed in natural populations

Upadhyay, Ranjit Kumar; Rai, Vikas

doi:10.1016/s0960-0779(97)00076-3

Cited by 81 publications

(75 citation statements)

References 5 publications

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“…This included the existence of stable focus, limit cycles and states of extinction. Moreover all simulations were performed for the mutual interference parameters m i ∈ [1,3]. This is well in accordance with our current results as we have relied on the compact Sobolev embedding…”

Section: Resultssupporting

confidence: 71%

“…This bound is immensely large, even for modest m i , say in the range [1,3]. It will be of much interest if we can sharpen this to perhaps the order of K or below.…”

Section: Resultsmentioning

confidence: 99%

“…THEOREM 2.1. Consider the diffusive three species aquatic model as defined via (1)- (3). For initial data in L 2 (Ω) there exists a unique weak solution (x 1 , x 2 , x 3 ) to the system such that…”

Section: The Mathematical Modelmentioning

confidence: 99%

“…He also took into account the realistic toxin liberation process of the TPP population. His work thus generalized several other well known models in the literature see [3] and [29]. However, Upadhyay's model as it stands, is an ODE model.…”

Section: Introductionmentioning

confidence: 99%

“…Why is the chaotic behavior of such systems although established theoretically, by dynamicists, not often observed in the wild, by ecologists [1], [47], [48]? Also see [5], [3], [47], [28] to this end. Thus there is a plethora of reasons why one might consider the study of food chain models to be both an intriguing and challenging intellectual pursuit.…”

Section: Introductionmentioning

confidence: 99%

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Investigation of the long time dynamics of a diffusive three species aquatic model

Parshad

Upadhyay

2010

Dynamics of Partial Differential Equations

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ABSTRACT. We consider Upadhay's three species aquatic model with the inclusion of spatial spread. We show the existence of aWe then derive uniform estimates to tackle the question of asymptotic compactness of the semi-group for the system in the Sobolev space. Via these we demonstrate the existence of a global attractor for the system which is compact in H 2 (Ω) and attracts all bounded sets in L 2 (Ω) in the H 2 (Ω) topology.

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Section: Resultssupporting

confidence: 71%

“…This bound is immensely large, even for modest m i , say in the range [1,3]. It will be of much interest if we can sharpen this to perhaps the order of K or below.…”

Section: Resultsmentioning

confidence: 99%

Section: The Mathematical Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Investigation of the long time dynamics of a diffusive three species aquatic model

Parshad

Upadhyay

2010

Dynamics of Partial Differential Equations

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A remark on “ Global dynamics of a tritrophic food chain model subject to the Allee effects in the prey population with sexually reproductive generalized‐type top predator” [Comp and Math Methods. 2019; E1079, pp. 1–23]

Takyi

Antwi-Fordjour

Kouachi³

et al. 2021

Comp and Math Methods

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In Debnath et al. (2019), a tritrophic food chain model subject to a Allee effect on the prey growth and Crowley-Martin senses functional response between intermediate predator and top predator, with a top predator of sexually reproductive type is considered. It is claimed that under certain restrictions on the parameter space, the model has bounded solutions for all positive initial conditions, and is dissipative. We show that this is not true. In particular, solutions to the model can blow-up in finite time, even under the restrictions derived in Debnath et al.(2019), for sufficiently chosen initial data. We derive a new extinction boundary for the system. We also conjecture on the effect of the Allee threshold on the blow-up dynamics in the model. All of our results are validated via numerical simulations. K E Y W O R D SAllee effect, finite time blow up, three species food chain

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Time delays and economic profit in fractional differential‐algebraic predator–prey model

Zhang,

Wang

2023

Math Methods in App Sciences

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In this paper, we formulate a fractional differential‐algebraic model with two discrete delays and double linear species harvesting. One of the delays denotes the time taken for digestion of the prey, and the other reflects the negative feedback of the predator's density. Taking into account of the economic profit, linear predator harvesting and prey harvesting have been incorporated into the proposed predator–prey system. From the biological viewpoint, we discuss the sufficient conditions for the existence of nontrivial positive equilibrium point. By jointly using the stability theory of fractional differential‐algebraic equations and the bifurcation theory, we study the delay‐induced instability and Hopf bifurcations. Finally, the correctness of the theoretical analysis is verified by numerical simulations, and the effects of delays, economic profit, and fractional exponents on the stability of the system are explored.

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Why chaos is rarely observed in natural populations

Cited by 81 publications

References 5 publications

Investigation of the long time dynamics of a diffusive three species aquatic model

Investigation of the long time dynamics of a diffusive three species aquatic model

A remark on “ Global dynamics of a tritrophic food chain model subject to the Allee effects in the prey population with sexually reproductive generalized‐type top predator” [Comp and Math Methods. 2019; E1079, pp. 1–23]

Time delays and economic profit in fractional differential‐algebraic predator–prey model

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