Stability switching and hydra effect in a predator–prey metapopulation model
Introduction
Over-exploitation of renewable resources is a main concern in conservation biology. It is one of the main threats for biodiversity (Wilcove et al., 1998) and many ecological fields such as fish, forest or wildlife are affected. Regulations have been developed to exploit resources in a sustainable manner. The concept of Maximum Sustainable Yield (MSY) emerged in the 1950’s for fisheries management (Schaefer, 1954, Schaefer, 1957) to define a maximum steady state harvest, meaning that the exploited population has no dynamics over the long term and then, does not collapse.
Many underlying processes occur while harvesting populations leading to drastic changes in what models predicted (Parma, 1998). First regulation studies focused on one-single species exploitation (e.g. Schaefer, 1954), however, interspecific interactions play a key role to understand how species exploitation affects their dynamics, and reversely. For instance, harvesting impacts trophic food webs, since such interactions tend to regulate population densities (Christensen, 1996). Further, industrial exploitation may lead to multi-species harvesting, so that both predator and prey can be harvested in the same time. This is why many studies have focused on predator–prey interactions (Brauer et al., 1976, Beddington and May, 1980).
The counterintuitive concept of “hydra effect”, coined after the Greek legend of the Lernaean Hydra which grew two heads for each one cut off, has recently received an increasing interest after the works of Abrams (2002). In ecological modelling, the hydra effect occurs when an increase of the death rate of a population results in an increase of the population size (Abrams and Quince, 2005). This concept has been widely shown to appear in predator–prey interactions, either in discrete-time models (Hilker and Westerhoff, 2006, Weide et al., 2019) or in continuous-time models (Matsuda and Abrams, 2004, Abrams and Quince, 2005, Sieber and Hilker, 2012). Predator or prey population may be subject to hydra effects if self-effects occur in one of the two species (Cortez and Abrams, 2016). Self-effects are due to positive density dependence occurring at equilibrium. For instance, Costa and dos Anjos (2018) have shown that, for a predator–prey system with Allee effects among predator, the hydra effects occur in predator population. Pal et al. (2019) have investigated food chains up to six-trophic level. No hydra effect at stable steady state is detected in a tritrophic food chain, but may appear in four-, five- and six-trophic food chains. da Silveira Costa (2008) established harvesting threshold policy to sustain populations when the predator population is harvested. Costa et al. (2017) have shown that the hydra effect occurs in stage-structure models. Recently, Ghosh et al. (2020) reported that the mature predator harvesting generates hydra effects in a significant range of effort.
Heterogeneous landscapes are recognized to deeply impact exploitation of populations. One of the modelling frameworks to represent population dynamics in a spatial environment is metapopulation (Levins, 1969), where population habitats are described by discrete patches connected by paths that individuals may use for their movements (Levin, 1974). Spatial heterogeneity comes from the difference between patches (i.e. habitats) leading to distinct demographic behaviours for populations living in these patches. Concerning predator–prey interactions, an interest for the effects of dispersal has grown in the 1990’s where stabilizing or destabilizing effects of dispersal have been studied. Kuang and Takeuchi (1994) have proposed a heterogeneous two-patch predator–prey model with prey migration and proved that the stable system can become unstable by regulating the migration rate. Jansen (1995) considered a two-patch Rosenzweig–MacArthur (RM) system and observed the asynchronous fluctuations of the local populations. Later, Jansen (2001) extended his results by considering the predator migration only. He found that (i) the oscillations in the spatial system always synchronize due to low migration rates, (ii) the intermediate migration rates produce chaotic attractors and (iii) for large predator migration rates may result the extinction of population in one of the patches.
The aim of the present paper is to study the interplay between the dispersal and harvesting of both the predator and prey. To do so, we consider a RM metapopulation model where both the prey and predator populations disperse and are harvested (Section 2). We analyse the general model in patches and then focus on the two-patch version (Section 3) to produce some qualitative results. In Section 4, we explore the impacts of dissimilar dispersal on the stability. Finally, we investigate the stability switching and hydra effect phenomenon in Section 5.
Section snippets
A metapopulation Rosenzweig–MacArthur model
Suppose a space is divided into patches. Let be the population vector of prey organized between patches and the population vector of predators. We consider that the total population vector is . The interactions between populations and in the patch are described as follows: A logistic growth describes the growth of the prey in absence of predators, where is the maximum growth rate
Model properties
In this section, we first prove the nonnegativity and boundedness of the model (2). Conditions of the local stability of the population-free equilibrium are then obtained. Finally, a parametric condition for the spatial heterogeneity is derived for the model (1) with two patches.
Let be the nonnegative quadrant and be the positive quadrant.
Proposition 3.1 [Nonnegativity of Solutions] The nonnegative quadrant remains invariant for the system (2).
Proof Suppose that, at a given time , one of the
Impact of dissimilar dispersal on stability
We investigate the impact of dispersal on the stability of the coexisting equilibrium, whenever it exists, in the model (1) with two patches. We assume that an intrinsic homogeneity occurs so that both the patches are identical in the absence of dispersal. Different sets of hypothetical parameters are chosen in order to obtain some possible qualitative outcomes on the dispersal induced stability. The parameter values are described in the caption of the Fig. 1. The stability changes due to the
Results on harvesting
In this section, we investigate the influence of the harvesting in the model (2) with two patches. We assume that both the patches are identical in the absence of dispersal and harvesting. The spatial heterogeneity of the model occurs by assuming different dispersal rates of either prey or predator between patches. Demographic parameter values are , , , , , , , . Numerically, the unharvested model with these parameters has a
Discussion and conclusion
In this study, a metapopulation model with predator–prey interactions is investigated to understand the dynamics of a well mixed system. We first established that the state variables of the general model are nonnegative and bounded. Then, the stability of the population-free equilibrium has been studied through linear algebra theory. The parametric condition to induce spatial heterogeneity into the unharvested two-patch model has also been derived.
We have numerically analysed the impacts of
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
N.B. acknowledges the support of the French Agence Nationale de la Recherche 664 (ANR) under grant ANR-13-BSV7-0011 (project FunFit) and of the SPE department of INRA, France (project ABCD). B.G. acknowledges SERB, Govt. of India for the financial support sanctioned under Early Carrier Research project (File No. ECR/2016/000677). N.B. is also thankful for the local financial assistant received from the same SERB project, India while visiting NIT Meghalaya for this collaborative work. We also
References (41)
- et al.
The impact of mortality on predator population size and stability in systems with stage-structured prey
Theor. Popul. Biol.
(2005) Spatio-temporal spread of infectious pathogens of humans
Infec. Dis. Model.
(2017)- et al.
Explicit impacts of harvesting in delayed predator-prey models
Chaos Solitons Fractals
(2019) - et al.
Maximum sustainable yields in systems subject to harvesting at more than one trophic level
Math. Biosci.
(1980) - et al.
Prey dynamics under generalist predator culling in stage structured models
Math. Biosci.
(2017) Predator harvesting in stage dependent predation models: Insights from a threshold management policy
Math. Biosci.
(2008)- et al.
Global stability for a predator-prey model with dispersal among patches
- et al.
Relationship between exploitation, oscillation, MSY and extinction
Math. Biosci.
(2014) - et al.
Harvesting induced stability and instability in a tri-trophic food chain
Math. Biosci.
(2018) The dynamics of two diffusively coupled predator–prey populations
Theor. Popul. Biol.
(2001)
The impact of immigration on a stability analysis of Lotka–Volterra system
IFAC PapersOnLine
Predator-prey dynamics in models of prey dispersal in two-patch environments
Math. Biosci.
Hydra effects in stable food chain models
Biosystems
Hydra effect and paradox of enrichment in discrete-time predator-prey models
Math. Biosci.
Will small population sizes warn us of impending extinctions?
Amer. Nat.
Diseases in metapopulations
Number of source patches required for population persistence in a source–sink metapopulation with explicit movement
Bull. Math. Biol.
Stabilization and de-stabilization of predator-prey systems under harvesting and nutrient enrichment
Internat. J. Control
Managing fisheries involving predator and prey species
Rev. Fish Biol. Fish.
Hydra effects in stable communities and their implications for system dynamics
Ecology
Cited by (10)
Persistence in spatial multi-species food webs: The conflicting influences of isolated food web feasibility and spatial asynchrony
2023, Communications in Nonlinear Science and Numerical SimulationEffects of intraguild prey dispersal driven by intraguild predator-avoidance on species coexistence
2022, Applied Mathematical ModellingUnderstanding the hydra effect in predator-dependent functional response models
2024, Discrete and Continuous Dynamical Systems - Series BFear and Group Defense Effect of a Holling Type IV Predator-Prey System Intraspecific Competition
2023, Emerging Science JournalNumerical treatment for mathematical model of farming awareness in crop pest management
2023, Frontiers in Applied Mathematics and StatisticsDesigning Meyer wavelet neural networks for the three-species food chain model
2023, AIMS Mathematics