Predator overcomes the Allee effect due to indirect prey–taxis
Introduction
The classical Rosenzweig – MacArthur model describing dynamics of a prey–predator system, assumes logistic growth of the prey population and the Holling type II functional response of the predator population (Rosenzweig and MacArthur, 1963). The well known property of this model consists in stability loss by the interior equilibrium in response to increase of parameters characterizing the voracity and/or fertility of the predator, and emergence of the limit cycle that becomes a global attractor of the system. The fact that amplitude of the cycle rapidly increases with further increase of the bifurcation parameter has led to conclusion about principal inability of the Rosenzweig – MacArthur model to reproduce the successful biological control of pest by potentially “effective”, i.e. voracious and rapidly reproducing agent species. Instead of expected durable suppression of the prey by its natural enemy the model demonstrates periodic outbreaks of the prey population, quickly followed by the predator abundance. Since basic hypotheses of the model, regarding reproduction of the prey and individual ration of the predator are supported by experiments on small-scale laboratory microcosms (see, e.g. Barlow, 1992, Bohannan, Lenski, 1997, Gauze, 1934, Haydon, Lloyd, 1999, Holling, 1959, Jeschke, Kopp, Tollrian, 2004, Kerfoot, DeMott, Levitan, 1985, Luckinbill, 1973, Tully, Cassey, Ferrière, 2005) i.e. they are empirically grounded, this contradiction between prediction of the theory and numerous observations on successful applications of biological method of pest and weed control in large-scale natural ecosystems (Beddington, Free, Lawton, 1978, Huffaker, 1957, Julien, Griffiths, 1998, Lenteren, Roermund, Sutterlin, 1996, McFadyen, 2000, Murdoch, Chesson, Chesson, 1985, Winder, Alexander, Holland, Symondson, Perry, Woolley, 2005) was recognized and named in the literature as “paradox of biological control” (Arditi, Berryman, 1991, Berryman, Hawkins, 1999, Luck, 1990).
In spatial models the paradox can naturally be resolved with explicit description of directed movements of predators, formalized as indirect prey–taxis (Sapoukhina et al., 2003), assuming that the taxis stimulus is a substance being continuously emitted by the prey, diffusing in space and decaying with constant rate in time (e.g. odour, pheromone, exometabolite) (Tyutyunov et al., 2017).
Such models with indirect prey–taxis can be equivalently reformulated in terms of inertial directed movements of predators, based on hypothesis that taxis acceleration is determined by the gradient of prey population density (Arditi, Tyutyunov, Morgulis, Govorukhin, Senina, 2001, Govorukhin, Morgulis, Tyutyunov, 2000, Morgulis, & Ilin, Tyutyunov, Titova, 2017). In contrast to conventional taxis–diffusion–reaction models effectively describing spatiotemporal dynamics of microbial and planktonic communities, with taxis velocity proportional to the prey gradient (e.g.Berezovskaya, Karev, 1999, Tsyganov, Biktashev, 2004, Tsyganov, Biktashev, Brindley, Holden, Ivanitsky, 2007, Tsyganov, Brindley, Holden, Biktashev, 2003, Tsyganov, Brindley, Holden, Biktashev, 2004, Wu, Shi, Wu, 2016), models with indirect (or inertial) prey–taxis capable to reproduce spatial clustering and heterogeneous wave regimes induced solely by spatial behaviour of predators belonging to highly developed taxa, without necessity of taking into account local population kinetics (i.e. predator’s reproduction and mortality processes) that occur at slower time scale comparing to animal movements (Arditi, Tyutyunov, Morgulis, Govorukhin, Senina, 2001, Tyutyunov, Zagrebneva, Surkov, Azovsky, 2009, Tyutyunov, Zagrebneva, Govorukhin, Titova, 2019). In particular, this approach to modelling directed movements of animals explains how successful application of biological method of pest control depends on spatial phenomena (Kovalev, Tyutyunov, 2014, Sapoukhina, Tyutyunov, Arditi, 2003, Tyutyunov, Kovalev, Titova, 2013a) as well as allows describing the pursuit–evasion phenomena, reproducing spatial pattern dynamics in prey and predator populations with conservative abundances (Tyutyunov, Titova, 2017, Tyutyunov, Titova, Arditi, 2007). Due to this and other advantages, this method for modelling animal taxis became quite popular during the last years (Arditi, Ginzburg, 2012, Chakraborty, Singh, Lucy, Ridland, 2007, Chakraborty, Singh, Lucy, Ridland, 2009a, Chakraborty, Singh, Ridland, 2009b, Govorukhin, Morgulis, Tyutyunov, 2000, Kuang, Ben-Arieh, Zhao, Wu, Margolies, Nechols, 2017, Morgulis, & Ilin, Rai, 2013, Rai, Upadhyay, Thakur, 2012, Sapoukhina, Tyutyunov, Arditi, 2003, Tello, Wrzosek, 2016, Thakur, Gupta, Upadhyay, 2017).
A limit cycle with large amplitude, which is periodically bringing either prey or predator population close to zero in the Rosenzweig – MacArthur model should be regarded as unrealistic per se, because such dynamics implies a high probability of population extinction due to either demographic or environmental inelasticity that are not taken into account by the model. As differential equations are incapable of describing growth of small population driven by a substantially stochastic process, adding the Allee effect (Allee, 1931, Stephens, Sutherland, 1999) makes the model predictions more naturalistic and cautious.
Allee effect consisting in a significant decrease in the per capita growth rate of a population at low population density, is a commonly observed feature of various species (Courchamp, Berec, Gascoigne, 2008, Courchamp, Clutton-Brock, Grenfell, 1999, Dennis, 1989, Stephens, Sutherland, 1999). The Allee effect can result in through various mechanisms, some of them are, namely, difficulties in finding mates, lowering in inbreeding rate, less number of successful mating, social facilitation of reproduction, aggression due to anti-predator behaviour, avoidance of predator due to evolutionary change, etc. (Allee, 1931, Banerjee, Takeuchi, 2017, Courchamp, Berec, Gascoigne, 2008, Courchamp, Clutton-Brock, Grenfell, 1999, Dennis, 1989, Stephens, Sutherland, 1999, Tyutyunov, Titova, Berdnikov, 2013b). Introduction of Allee effect in single species population growth does not alter the resulting dynamics significantly when populations are assumed to be homogeneously distributed over their habitat and any kind of stochasticity is ignored except the possibility of extinction of species if their density goes below some threshold (Amarasekare, 1998, Boukal, Berec, 2002, Boukal, Sabelis, Berec, 2007, Dennis, 2002, Jankovic, Petrovskii, 2014, Morozov, Banerjee, Petrovskii, 2016, Petrovskii, Blackshaw, Li, 2008). A wide variety of two species prey–predator models with Allee effect in prey growth are proposed and analyzed (see Aguirre, González-Olivares, Sáez, 2009a, Aguirre, González-Olivares, Sáez, 2009b, Berec, Angulo, Courchamp, 2007, González-Olivares, Mena-Lorca, Rojas-Palma, Flores, 2011, Sen, Banerjee, 2015, Sen, Banerjee, Morozov, 2012, Stephens, Sutherland, 1999, Van Voorn, Hemerik, Boer, Kooi, 2007, Zhou, Liu, Wang, 2005 and references therein) to explain how the lowering of the prey growth, when their population density is low, affects the dynamics of the system and sometime leads to total extinction of both species. Various mathematical forms are proposed to capture different types of the Allee effect and concise description is provided in Courchamp et al. (2008). On the other hand the two species prey–predator type models with Allee effect only in predator growth are rare in literature, with some exceptions including recent works by Wang et al. (2013), Bodine and Yust (2017), and by Alves and Hilker (2017). In the later paper Alves and Hilker (2017), the Allee effect in predator growth is induced by the strength of hunting cooperation among the predators. It is quite reasonable to assume that the resource for specialist predator is abundant but they suffer from lowering in their growth rate due to less probability of successful mating when their density is low. Based upon this point here we propose a prey–predator model, predator is specialist in nature, with Allee effect in predator growth.
In particular, the main spatiotemporal model is aimed at answering a question whether the prey–taxis ability gives additional advantages to the predator with the Allee effect. Earlier it has been demonstrated that due to prey–taxis activity, predators increase their individual ration and fecundity and at the same time preserve the prey from local depletion (Arditi, Tyutyunov, Morgulis, Govorukhin, Senina, 2001, Govorukhin, Morgulis, Tyutyunov, 2000, Sapoukhina, Tyutyunov, Arditi, 2003, Tyutyunov, Sapoukhina, Senina, Arditi, 2002, Tyutyunov, Sapukhina, Morgulis, Govorukhin, 2001).
In the present work we study a prey–taxis model based on non-spatial Rosenzweig – MacArthur prey–predator system (Rosenzweig and MacArthur, 1963), with Allee effect in predator population growth (Zhou et al., 2005), assuming that efficiency of the predator reproduction decreases at low population abundances. Results of analytical and simulation studies help better understand the role of the predator’s spatial activity in population viability and species fitness.
This paper is organized as follows. After presenting a general taxis–diffusion–reaction system, we first investigate dynamic properties of a point, i.e. non-spatial case of the model describing local kinetics of the considered trophic system. Next, we perform detailed study of the full spatial model, giving results of the linear analysis of stability of homogeneous stationary state of the model with respect to small spatially heterogeneous perturbations. Numerical simulations illustrate and verify analytical results. Stability of homogeneous periodic regime is also studied with help of simulations. In the discussion section special attention is paid to biological interpretation and theoretical consequences of results with regard to earlier findings obtained with prey–taxis models not taking into account the Allee effect, and with diffusive models that include the Allee effects.
Section snippets
Model
We consider a spatial prey–predator model with specialist predator and the growth of predator population subjected to the Allee effect. For the case of simplicity of the model presentation it is assumed that the habitat of the trophic system is a one-dimensional domain . Thus the prey and predator populations are represented by their densities, and respectively, defined at time t and spatial position x ∈ [0, L]. In absence of the Allee effect the local prey–predator
Non-spatial case of the model
The proposed spatial model (3)–(4) is a three-component system. However in spatially-homogeneous case the model is being reduced to a two-component prey–predator system. Namely, with spatially-homogeneous initial conditions all terms containing spatial derivatives in system (3) vanish, and dynamics of both prey and predator populations becomes independent on stimulus S. Thus the non-spatial case of model (3)–(4) is represented by the following coupled nonlinear ordinary differential equations:
Discussion and conclusion
We have considered a spatial model of a prey–predator system, performing linear stability analysis and numerical simulations for both spatially homogeneous and heterogeneous cases of population dynamics. The model takes into account two phenomena common for many natural populations of predatory species: the Allee effect in predator population growth, and indirect prey–taxis that formalizes the predator’s capability of sensing the environmental cues related to spatial distribution of prey. The
Acknowledgements
First author, Yuri Tyutyunov, acknowledges funding by the research project 0256-2019-0038 (state reg. no. 01201363188) of the Southern Scientific Centre of the Russian Academy of Sciences (SSC RAS) “Development of GIS-based methods of modelling marine and terrestrial ecosystems”, and RFBR grant 18-01-00453 “Multistable spatiotemporal scenarios for population models”. Second author, Deeptajyoti Sen, acknowledges funding supported by the University Grant Commission, India. Third author, Lyudmila
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