Additional food supplements as a tool for biological conservation of predator-prey systems involving type III functional response: A qualitative and quantitative investigation

Pappula, Srinivasu; Vamsi, D. K. K.; Ananth, V.S.

doi:10.1016/j.jtbi.2018.07.019

Cited by 24 publications

(35 citation statements)

References 25 publications

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“…Not limited to this problem, mathematical modelling of population dynamics in natural environment is a heart of ecological management [52]. Recent examples include invasive species management [28], tracking predator-prey dynamics for biological conservation [50], and sustainable fisheries management [19]. The population dynamics is often inherently stochastic mainly due to unresolved nonlinearities involved in the ecological and environmental processes [31].…”

Section: Introductionmentioning

confidence: 99%

A simplified stochastic optimization model for logistic dynamics with control-dependent carrying capacity

Yoshioka

2019

Journal of Biological Dynamics

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A simplified stochastic control model for optimization of logistic dynamics with the control-dependent carrying capacity, which is motivated by a recent algae population management problem in the river environment, is presented. Solving the optimization problem reduces to finding a solution to a non-local first-order differential equation called tt-Jacobi-Bellman (HJB) equation. It is shown that the HJB equation has a unique viscosity solution and that the solution can be approximated with a finite difference scheme. Asymptotic estimates of the solution and the optimal control are derived and compared with numerical solutions. Finally, parameter dependence of the optimal control is examined numerically with implications to river environmental management.

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Section: Introductionmentioning

confidence: 99%

A simplified stochastic optimization model for logistic dynamics with control-dependent carrying capacity

Yoshioka

2019

Journal of Biological Dynamics

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“…These natural enemies are provided with additional food supplements, which affect their survival, gustatory response, fitness, fecundity and longevity Landis et al (2000) ; Urbaneja-Bernat et al (2013) ; Winkler et al (2005) and this in turn helps in controlling pests in the system. Some of the outcomes of the mathematical studies of additional food systems Das and Samanta, 2018a , Das and Samanta, 2018b , Das and Samanta, 2020 ; Liu et al (2018) ; Mondal and Samanta, 2019 , Mondal and Samanta, 2020 ; Prasad and Prasad (2019) ; Sahoo and Poria (2014) ; Sen et al (2015) ; Srinivasu et al, 2007 , Srinivasu et al, 2018 ; Vamsi et al (2019) reveal that the provision of additional food to the predators affects the global dynamics of the system causing indirect interaction of species. These findings are in line with the experimental observations when additional food is provided Harwood et al (2004) ; Wade et al (2008) ; Toft (2005) .…”

Section: Introductionmentioning

confidence: 99%

“…This is in contrast with the behaviour of systems involving type III response where rate of predation depends on the prey density. This is because the type III response depends on the prey detectability nature of the predator Srinivasu et al (2018) .…”

Section: Introductionmentioning

confidence: 99%

“…Recently, in the work Srinivasu et al (2018) , the authors have modeled and studied the following additional food provided prey-predator system involving type III functional response

where N denotes the prey density, P the predator density and A the biomass of the additional food which is provided to the predators. Similar to the mathematical works mentioned above, we assume that P represents generalist predators which can feed on different diet supplements other than their target prey.…”

Section: Introductionmentioning

confidence: 99%

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Influence of quantity of additional food in achieving biological conservation and pest management in minimum-time for prey-predator systems involving Holling type III response

Ananth

Vamsi

2021

Heliyon

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Incorporating additional food supplements into the predators' diet complementary to the target prey has gained importance over the years due to its pertinence in achieving biological conservation and biological control. Studies by theoretical ecologists and mathematicians reveal that by providing appropriate quality and quantity of additional food to the predator, the system could be driven either towards co-existence of species (to an admissible interior equilibrium), thereby achieving conservation or towards elimination of either of species achieving bio-control eventually with time. However, one of the limitations of these studies is that the desired state is reached only as asymptotes which makes the outcomes of the studies not that practically viable. In this work, to overcome the limitation of asymptotes, we formulate and study a time optimal control problem for additional food provided system involving type III response using quantity of additional food as the control. The objective of the study is to reach the desired terminal state in minimum time. To that end, we first prove the existence of optimal solution using the Filippov's existence theorem and then establish the characteristics of the optimal control using the Pontryagin's Maximum Principle . Using the Hamiltonian minimization condition and the monotonicity property of the Hamiltonian with respect to the quantity parameter, we show that the optimal control strategy is of bang-bang type with a possibility of multiple switches in the trajectory in case of biological conservation and no switch in case of pest management. Since the additional food system exhibits contrasting behaviour with respect to quality additional food, we have considered multiple cases of quality as a part of this study and in each case, we fixed the quality parameter as constant. The theoretical results have been illustrated by performing numerical simulations for various cases relating to both biological conservation and pest management. The theoretical outcomes of this study are in line with ecological field observations.

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“…If ϵ ∈ (0, 1) represents the efficiency with which the food consumed by the predator gets converted into predator biomass then the maximum growth rate of the predator given by ϵc 1 is represented by the parameter b 1 , and therefore, the growth rate of predator P is of the form f (N ) = b 1 (N +ξ) a 1 +αξ+N . Incorporating the above facts, Kumar and Chakrabarty [12], Srinivasu et al [13], Sahoo and Poria [14] and Ghorai and Poria [15] proposed and analyzed the following predator-prey system with additional food…”

Section: Introductionmentioning

confidence: 99%

Extinction and ergodic stationary distribution of a Markovian-switching prey-predator model with additional food for predator

Guo

Ruan

2020

Math. Model. Nat. Phenom.

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In this work we have studied a stochastic predator-prey model where the prey grows logistically in the absence of predator. All parameters but carry ing capacity have been perturbed with telephone noise. The prey’s growth rate and the predator’s death rate have also been perturbed with white noises. Both of these noises have been proved extremely useful to model rapidly fluctuating phenomena Dimentberg (1988). The conditions under which extinction of predator and prey populations occur have been established. We also give sufficient conditions for positive recurrence and the existence of an ergodic stationary distribution of the positive solution, which in stochastic predator-prey systems means that the predator and prey populations can be persistent, that is to say, the predator and prey populations can be sustain a quantity that is neither too much nor too little. In our analysis, it is found that the environmental noise plays an important role in extinction as well as coexistence of prey and predator populations. It is shown in numerical simulation that larger white noise intensity will lead to the extinction of the population, while telephone noise may delay or reduce the risk of species extinction.

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Additional food supplements as a tool for biological conservation of predator-prey systems involving type III functional response: A qualitative and quantitative investigation

Cited by 24 publications

References 25 publications

A simplified stochastic optimization model for logistic dynamics with control-dependent carrying capacity

A simplified stochastic optimization model for logistic dynamics with control-dependent carrying capacity

Influence of quantity of additional food in achieving biological conservation and pest management in minimum-time for prey-predator systems involving Holling type III response

Extinction and ergodic stationary distribution of a Markovian-switching prey-predator model with additional food for predator

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