Uniform Persistence and Periodic Solutions of Generalized Predator–Prey Type Eco-Epidemiological Systems

2021 ◽  
Vol 31 (02) ◽  
pp. 2150033
Author(s):  
Mengfeng Sun ◽  
Guoting Chen ◽  
Xinchu Fu
Keyword(s):  
Functional Response ◽  
Allee Effect ◽  
Three Dimensional ◽  
Prey Type ◽  
Uniform Persistence ◽  
Predator Population ◽  
Allee Effects ◽  
Biologically Relevant ◽  
Two Factors ◽  
Nonmonotonic Functional Response

In this paper, we analyze a class of three-dimensional eco-epidemiological models where prey is subject to Allee effects and infection. We first establish the existence, uniqueness, positivity and uniform ultimate boundedness of the solutions for the proposed system in the positive octant. For three subsystems, we investigate the existence of their respective trivial and positive equilibria and determine the conditions for some bifurcations (Hopf bifurcation, Bogdanov–Takens bifurcation of codimension-2 and saddle-node bifurcation) to occur. We find that the Allee effect, nonmonotonic functional response and intra-class competition in susceptible preys enable the S–I and S–P subsystems to have richer dynamics. For example, the S–I subsystem can have up to three positive equilibria, the S–P subsystem with nonmonotonic functional response can have two positive equilibria while it is impossible in monotonic situation, and high intra-class competition in susceptible preys may lead to the extinction of the predator population, etc. We show that the strong Allee effect can create a separatrix curve (or surface), leading to multistability. Then, we study the uniform persistence of the full system and identify an interior periodic orbit by applying Poincaré map and bifurcation theory. Our analysis reveals that the introduction of the infection or predation may act as a biological control to save the population from extinction and the interaction between these two factors yields a diverse array of biologically relevant behaviors. Finally, some numerical simulations are performed to support and supplement our analytical findings.

scholarly journals Dynamics of prey–predator model with strong and weak Allee effect in the prey with gestation delay

2020 ◽  
Vol 25 (3) ◽  
Author(s):  
Ankit Kumar ◽  
Balram Dubey
Keyword(s):  
Allee Effect ◽  
Threshold Value ◽  
Prey Population ◽  
Predator Population ◽  
Allee Effects ◽  
Normal Form Theory ◽  
Delayed Systems ◽  
Gestation Delay ◽  
The Status ◽  
Manifold Theory

This study proposes two prey–predator models with strong and weak Allee effects in prey population with Crowley–Martin functional response. Further, gestation delay of the predator population is introduced in both the models. We discussed the boundedness, local stability and Hopf-bifurcation of both nondelayed and delayed systems. The stability and direction of Hopfbifurcation is also analyzed by using Normal form theory and Center manifold theory. It is shown that species in the model with strong Allee effect become extinct beyond a threshold value of Allee parameter at low density of prey population, whereas species never become extinct in weak Allee effect if they are initially present. It is also shown that gestation delay is unable to avoiding the status of extinction. Lastly, numerical simulation is conducted to verify the theoretical findings. 

Emergent Allee Effects through Biomass Overcompensation

2013 ◽  
Author(s):  
André M. de Roos ◽  
Lennart Persson
Keyword(s):  
Growth Rate ◽  
Population Growth ◽  
Positive Feedback ◽  
Population Growth Rate ◽  
Allee Effect ◽  
Prey Availability ◽  
Population Level ◽  
Life History Stage ◽  
Predator Population ◽  
Allee Effects

This chapter discusses the emergence of a positive feedback between the density of predators and the availability of its food, mediated through biomass overcompensation in the prey life history stage that it forages on. This positive feedback between predation, prey availability, and thus predator population growth rate manifests itself at the population-level as an Allee effect for the predator: a predator population at low density will decline to extinction, whereas at high densities predators will manage to establish themselves in a community with prey. However, this positive relation between predator density and its population growth rate does not result from any positively density-dependent interactions among the predators themselves, which generally form the basis of an Allee effect. Instead, predators only interact with each other through exploitative competition for prey. The Allee effect emerges solely as a consequence of the demographic changes in the prey population, which are induced by the mortality that the predator imposes. For this reason this phenomenon is referred to as an “emergent Allee effect.”

scholarly journals A Modified Leslie–Gower Model Incorporating Beddington–DeAngelis Functional Response, Double Allee Effect and Memory Effect

2021 ◽  
Vol 5 (3) ◽  
pp. 84
Author(s):  
Emli Rahmi ◽  
Isnani Darti ◽  
Agus Suryanto ◽  
Trisilowati
Keyword(s):  
Functional Response ◽  
Memory Effect ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Predator Population ◽  
Predator Prey Model ◽  
Proposed Model ◽  
Caputo Fractional Order Derivative ◽  
Extinction Point

In this paper, a modified Leslie–Gower predator-prey model with Beddington–DeAngelis functional response and double Allee effect in the growth rate of a predator population is proposed. In order to consider memory effect on the proposed model, we employ the Caputo fractional-order derivative. We investigate the dynamic behaviors of the proposed model for both strong and weak Allee effect cases. The existence, uniqueness, non-negativity, and boundedness of the solution are discussed. Then, we determine the existing condition and local stability analysis of all possible equilibrium points. Necessary conditions for the existence of the Hopf bifurcation driven by the order of the fractional derivative are also determined analytically. Furthermore, by choosing a suitable Lyapunov function, we derive the sufficient conditions to ensure the global asymptotic stability for the predator extinction point for the strong Allee effect case as well as for the prey extinction point and the interior point for the weak Allee effect case. Finally, numerical simulations are shown to confirm the theoretical results and can explore more dynamical behaviors of the system, such as the bi-stability and forward bifurcation.

scholarly journals Publisher Correction: Predicting Subtype Selectivity for Adenosine Receptor Ligands with Three-Dimensional Biologically Relevant Spectrum (BRS-3D)

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Song-Bing He ◽  
Ben Hu ◽  
Zheng-Kun Kuang ◽  
Dong Wang ◽  
De-Xin Kong
Keyword(s):  
Adenosine Receptor ◽  
Three Dimensional ◽  
Receptor Ligands ◽  
Biologically Relevant ◽  
Subtype Selectivity ◽  
Adenosine Receptor Ligands

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

scholarly journals The influence of density in population dynamics with strong and weak Allee effect

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
Kamrun Nahar Keya ◽  
Md. Kamrujjaman ◽  
Md. Shafiqul Islam
Keyword(s):  
Population Dynamics ◽  
Allee Effect ◽  
Global Bifurcation ◽  
Reaction Diffusion ◽  
Logistic Growth ◽  
Allee Effects ◽  
Reaction Diffusion Model ◽  
Positive Steady States ◽  
The Stability ◽  
The Impact

AbstractIn this paper, we consider a reaction–diffusion model in population dynamics and study the impact of different types of Allee effects with logistic growth in the heterogeneous closed region. For strong Allee effects, usually, species unconditionally die out and an extinction-survival situation occurs when the effect is weak according to the resource and sparse functions. In particular, we study the impact of the multiplicative Allee effect in classical diffusion when the sparsity is either positive or negative. Negative sparsity implies a weak Allee effect, and the population survives in some domain and diverges otherwise. Positive sparsity gives a strong Allee effect, and the population extinct without any condition. The influence of Allee effects on the existence and persistence of positive steady states as well as global bifurcation diagrams is presented. The method of sub-super solutions is used for analyzing equations. The stability conditions and the region of positive solutions (multiple solutions may exist) are presented. When the diffusion is absent, we consider the model with and without harvesting, which are initial value problems (IVPs) and study the local stability analysis and present bifurcation analysis. We present a number of numerical examples to verify analytical results.

scholarly journals A cryptic Allee effect: spatial contexts mask an existing fitness–density relationship

2015 ◽  
Vol 2 (6) ◽  
pp. 150034 ◽  
Author(s):  
Akira Terui ◽  
Yusuke Miyazaki ◽  
Akira Yoshioka ◽  
Shin-ichiro S. Matsuzaki
Keyword(s):  
Allee Effect ◽  
Local Level ◽  
Freshwater Mussel ◽  
Fertilization Rate ◽  
Positive Density ◽  
Allee Effects ◽  
Flow Conditions ◽  
Conspecific Density ◽  
Sperm Density ◽  
Density Relationship

Current theories predict that Allee effects should be widespread in nature, but there is little consistency in empirical findings. We hypothesized that this gap can arise from ignoring spatial contexts (i.e. spatial scale and heterogeneity) that potentially mask an existing fitness–density relationship: a ‘cryptic’ Allee effect. To test this hypothesis, we analysed how spatial contexts interacted with conspecific density to influence the fertilization rate of the freshwater mussel Margaritifera laevis . This sessile organism has a simple fertilization process whereby females filter sperm from the water column; this system enabled us to readily assess the interaction between conspecific density and spatial heterogeneity (e.g. flow conditions) at multiple spatial levels. Our findings were twofold. First, positive density-dependence in fertilization was undetectable at a population scale (approx. less than 50.5 m 2 ), probably reflecting the exponential decay of sperm density with distance from the sperm source. Second, the Allee effect was confirmed at a local level (0.25 m 2 ), but only when certain flow conditions were met (slow current velocity and shallow water depth). These results suggest that spatial contexts can mask existing Allee effects.

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