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Allee effect in a prey–predator system
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Chaos, Solitons and Fractals 36 (2008) 334–342 www.elsevier.com/locate/chaos
Allee effect in a prey–predator system Despina Hadjiavgousti, Simos Ichtiaroglou
*
Department of Physics, University of Thessaloniki, 54124, Greece Accepted 19 June 2006
Abstract We construct a model of a specialist predator and its prey species for the case of non-overlapping generations and we assume that the prey obeys to the Allee effect. We observe that, for small values of the minimal viable population of the prey, there are stable dynamical configurations corresponding either to stable fixed points or to a stable limit cycle which supports quasiperiodic orbits. In this last case we observe the paradox of enrichment, where an initial increase of the population of the prey may lead to cycles of very fast increase and subsequent decrease of the predator population. This behavior increases the probability of extinction due to unpredictable external factors. We consider also non-zero dispersal probability of the prey to neighboring sites and observe that predation causes the disappearance of a rescue effect, even when the Allee threshold is small. 2006 Elsevier Ltd. All rights reserved.
1. Introduction The reduction of the per capita growth rate of a population of a biological species at densities smaller than a critical value is known as the Allee effect, see e.g. [1–5]. The main cause of the Allee effect is the difficulty in finding mates between the individuals of a species at low population densities. Other causes may be reduced defense against predators, special trends of social behavior, etc. The dynamical description by difference equations (maps) is valid in the case of populations with distinct, non-overlapping generations, for example some insect populations. Such a dynamical description has often been used in population dynamics in connection to logistic type maps, e.g. [6,7]. In two previous papers [8,9] a number of identical sites, appropriate for colonization by a biological species obeying the Allee effect, was considered. It was assumed that initially the central site carries a population equal to the carrying capacity of the environment, while all other sites are empty. For non-zero dispersal probability three final configurations were observed: (a) the source-sink effect, where almost all individuals remain at the central site, while the populations at the nearby sites attain very small values, significantly smaller than the Allee threshold, (b) the rescue effect, where complete colonization at all sites occurs through dispersal from the central one and (c) the extinction of the metapopulation for relatively large values of the dispersal probability. In [8] the Allee effect was modeled by a third degree polynomial map, while a polynomial of fifth degree was used in [9]. In the latter case, there was a control of chaotic behavior by increasing the dispersal probability.
*
Corresponding author. E-mail address: [email protected] (S. Ichtiaroglou).
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.053
D. Hadjiavgousti, S. Ichtiaroglou / Chaos, Solitons and Fractals 36 (2008) 334–342
335
In the present paper, in order to study the combined influence of predation and the Allee effect on a population, we construct a prey–predator model, compatible with the following assumptions: (a) both populations have non-overlapping generations, so the dynamics is described by a discrete map, (b) the dynamics of the prey, in absence of predator, is subjected to the Allee effect, (c) the predator is specialist, see e.g. [10–13] and its dynamics is of logistic type, (d) the normalized value (1, 1) is a fixed point of the map and (e) the square U = [0, 1] · [0, 1] remains invariant under the map. We study the fixed points and their stability and obtain phase portraits for various values of the parameters. We observe that, for small values of the Allee threshold, i.e. the minimal viable population of the prey, there are stable dynamical configurations corresponding either to stable fixed points or to a limit cycle which supports quasiperiodic orbits. This latter behavior relates to the paradox of enrichment [14–20], where an initial increase of the prey population leads to cycles of very fast increase and subsequent decrease (‘‘boom and bust’’, [21]) of the predator population, and this fact increases the probability of extinction of the predator due to unpredictable external factors. In contrast, for relatively large values of the Allee threshold or the predation parameter, both populations are led to extinction. Finally, we consider a chain of five identical sites, suitable for colonization by the prey species and non-zero probability of dispersal from the central site to the neighboring, initially empty sites. We observe that, even for small values of the minimal viable population of the prey, predation prohibits the appearance of a rescue effect, in accordance to other models, where predation may slow, stop or even reverse the invasion of a prey [22,23] or create a patchy distribution of its population [24,25].
2. The model We introduce a mapping model for prey–predator populations localized at one site. We consider the map xnþ1 ¼ F 1 ðxn ; y n Þ ¼ xn xn ð1 xn Þða xn Þ bxn y n ð1 xn Þ; xn y n ð3 y n Þ ; y nþ1 ¼ F 2 ðxn ; y n Þ ¼ 1 þ xn
ð1Þ ð2Þ
where x, y 2 U = [0, 1] · [0, 1], Fi : U ! U, i = 1, 2, i.e. the unit square U remains invariant. The variables xn, yn represent, respectively, the normalized populations of a prey and its predator at the nth generation. The functions F1, F2 indicate the relation between the reproduction and death rates of the two populations and the predation rate at a certain localized site, appropriate for colonization. The term xng1(xn) = xn[1 (1 xn)(a xn)] stands for the rate of the increase of the prey population in absence of predator and indicates that the prey obeys to the Allee effect. If it is smaller than a critical value x = a < 1, it will extinct. The parameter a is the Allee threshold and equals to the minimal viable population of the prey. The term xng2(xn, yn) = yn[bxn(1 xn)] represents the rate of decrease due to predation. This rate is proportional to the density yn and the predation ability g2 of the predator. This ability ascribes the relative predation rate, i.e. the number of individuals killed by the predator at one iteration of the map. The parameter b is the predation parameter. In this paper we choose 0.1 < b < 0.8. The dynamics of the prey in absence of predation (b = 0) has been studied in detail in Ref. [8]. The term yng3(xn, yn) = ynxn(3 yn)/(1 + xn) stands for the variation of the predator density with respect to the prey population. Note that, if the prey disappears, the population of the predator extincts at one generation. The predator is specialist, i.e. it feeds on this particular prey only, so that the point (0, 0) is an asymptotically stable fixed point and corresponds to the extinction of both populations. In addition, we assume the dynamics of the predator being of the logistic type and (1, 1) is also a fixed point of the map.
3. Fixed points and stability The fixed points of the map (2) are determined as the solutions of the system x = F1(x, y), y = F2(x, y) which belong in U. Let ! oF 1 oF 1 DF ¼
ox
oy
oF 2 ox
oF 2 oy
be the matrix of the linearized map at a fixed point. This point is a stable (unstable) node if both eigenvalues of DF are real and absolutely smaller (greater) than 1. If the eigenvalues are real and one is smaller while the other is greater than 1, it is a saddle point. If the eigenvalues are complex conjugates with real part smaller (greater) than 1, the point is a stable (unstable) focus. The map (2) has the following fixed points:
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• (x, y) = (0, 0): The eigenvalues are 1 a, 0, so the point is a stable node. • (x, y) = (1, 0): The eigenvalues are a, 3/2 and the point is a saddle. The axis y = 0 and the line x = 1 are the stable and unstable manifolds, respectively. • (x, y) = (1, 1): The eigenvalues are a + b, 1/2, so the point is a stable node for a + b < 1. For a + b > 1 it is a saddle and the lines x = 1 and x = y are the stable and unstable manifold, respectively. • (x, y) = (a, 0): The eigenvalues are 1 + a(1 a), 3a/(1 + a). The point is a saddle for a < 0.5 with the axis y = 0 the unstable manifold. The stable manifold depends on b. For a > 0.5 it is an unstable node. • (x, y) = (x*, y*) where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ a þ by ; y ¼ ½ð2b aÞ ð2b aÞ2 þ 4bð1 2aÞ=2b: These points are taken into account when they are real and belong to U. Their stability will be studied through the phases portraits of the map.
4. Dynamics of the map We will study numerically the map (2) for two values of a, one smaller and one greater than the bifurcation value 0.5, and for several values of b. These cases correspond to a small and a large value of the minimal viable population of the prey, i.e. the Allee threshold, respectively. In the second case the Allee effect is expected to manifest itself more strongly. We select the values a = 0.3 and a = 0.6. In all figures the stable nodes are represented by circles, the unstable node by a square, and the saddles by rhombuses. In Fig. 1, the phase portraits of the map (2) for a = 0.3 are shown. For this value of a, the fixed point at (a, 0) is a saddle for all values of b and the state space is partitioned in two domains by the separatrix which coincides with the stable manifold of this point. All the initial conditions lying in the domain at the left side, i.e. corresponding to small initial values of the prey population, lead to the stable node at (0, 0), i.e. to extinction of both populations. This domain is the basin of attraction of (0, 0), where the influence of the Allee effect for small x is evident. For b < 0.666, the domain at the right of the separatrix is the basin of attraction of the stable node at (1, 1) (Fig. 1a and b). Thus, for small values of the predation parameter, the populations are lead either to extinction or to the carrying capacity of the environment, depending on whether the prey population is initially small or not. We observe also that, as the predation parameter increases, the separatrix bends, resulting to an increase of the extinction domain. For b = 0.666 a saddle-focus bifurcation occurs and the two fixed points (x*, y*) appear, as can be seen at Fig. 1c for b = 0.68. The final state of the system, besides extinction, may be either the point (1, 1) or the stable focus. For b = 0.7 (Fig. 1d) we have a + b = 1 and a subcritical bifurcation occurs. The saddle point generated by the saddle-focus bifurcation collides with the stable node at (1, 1) interchanging stability. The point (1, 1) becomes a saddle and the stable node moves out of the square U. At the same time, the stable focus undergoes a Hopf bifurcation becoming unstable and a stable invariant limit cycle, supporting quasiperiodic orbits, appears (Fig. 1e). Increasing b further, the invariant circle increases in size. Thus, it is possible for the system to perform stable quasiperiodic oscillations, during which, while the prey population remains sufficiently large and the Allee effect is not operating, the predator population attains extremely small values. Although, in this model the predator does not extinct during the oscillations, in a realistic situation these small values may put a threat of extinction due to unpredictable external factors. This behavior corresponds to the paradox of enrichment, where an initially increased prey population leads to cycles of very fast increase and subsequent decrease of the predator population. As mentioned in the introduction, this fact increases the probability of extinction of the predator due to unpredictable external factors. This behavior is maintained up to b ’ 0.798, where the invariant circle touches the stable manifolds of the saddles at (a, 0) and (1, 1) and disappears. At the same time, a heteroclinic connection is formed between these two saddle points. For still larger values of b, the stable manifold of (a, 0) winds around the unstable focus, while the unstable manifold of (1, 1) is connected to the stable node at (0, 0). This shift of the invariant manifolds by varying b is shown in Fig. 2. For values of the predation parameter b > 0.798, the node at (0, 0) is the only attracting fixed point, so all initial conditions in the unit square U lead to extinction, with the exception of the points on the stable manifold of (a, 0), which are of measure zero. As is apparent from the spiraling orbits in Fig. 1f, both populations may attain significantly large values, before ultimate extinction. In Fig. 3, the phase portraits of system (2) for a = 0.6 are shown. For a > 0.5 and a + b < 1, there is always only one fixed point (x*, y*) 2 U, which is a saddle. So, in Fig. 3a for b = 0.3, the stable configurations correspond to the points
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Fig. 1. The state space for a = 0.3 and b = 0.1 (a), b = 0.6 (b), b = 0.68 (c), b = 0.7 (d), b = 0.72 (e), b = 0.8 (f).
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Fig. 2. The asymptotic manifolds for (a) b = 0.72, (b) b = 0.798 (the heteroclinic connection) and (c) b = 0.85.
Fig. 3. The state space for a = 0.6 and b = 0.3 (a) and b = 0.5 (b).
(0, 0) and (1, 1). Most of the initial conditions lead to extinction, but a set of initial conditions for large values of the prey population, leads to the normalized value (1, 1) which is the carrying capacity of the environment. Comparing this behavior to the one shown in Fig. 1a and b, we observe that, if the Allee effect manifests itself strongly (large a), a large population of the prey is demanded for survival of both populations. For a + b = 1, a subcritical bifurcation occurs and the saddle (x*, y*) collides with the stable node at (1, 1) and interchange their stability. Thus, for values of b > 0.4 the only stable configuration in U is the node at (0, 0) and all initial conditions lead to extinction, as is shown in Fig. 3b for b = 0.5. In this case too, for large values of prey, the population of the predator may attain significant values, before it extincts.
5. Patchy distribution of the prey In this section, we consider a chain consisting of five identical sites, which are appropriate for colonization by the prey species, while the predator explores all of them and predates on the whole metapopulation of the prey at all sites with equal probability. We define each site by an integer m, such that 2 6 m 6 2 and, at time n, the prey population at this site is xnðmÞ . Each individual of the prey either emigrates to the nearest neighbor sites to the right or left, with equal probability p, or remains at the same site, with probability 1 2p. The probability of dispersal to a non-nearest neighboring site equals to zero. An individual at the boundary sites of the chain m = ±2, may either move to the site m = ±1, or remain at the same site, or escape from the population. The mapping for this system is described by the following equations:
D. Hadjiavgousti, S. Ichtiaroglou / Chaos, Solitons and Fractals 36 (2008) 334–342 ðmÞ
xnþ1 ¼ pðxðmþ1Þ þ xðm1Þ Þ þ ð1 2pÞF 1 ðxðmÞ n n n ; y n Þ; 2 P
y n ð3 y n Þ 1þ
2 P
ð3Þ
xnðmÞ
m¼2
y nþ1 ¼
339
ð4Þ
ðmÞ
xn
m¼2
where ðmÞ ðmÞ ðmÞ ðmÞ ðmÞ ðmÞ F 1 ðxðmÞ n ; y n Þ ¼ xn xn ð1 xn Þða xn Þ bxn y n ð1 xn Þ:
Initially, at n = 0, we consider that the population of the prey at the central site m = 0 equals the carrying capacity of ð0Þ ðmÞ the site in absence of the predator, x0 ¼ 1, while all other sites are empty, x0 ¼ 0, m 5 0. The initial value of the predator population is also considered to be y0 = 1. In Ref. [8] we studied the behavior of the above model in absence of predation, i.e. for b = 0 and y = 0 for the above initial conditions. It was shown that, for sufficiently small values of p there is a localization of the metapopulation at the central site, which corresponds to the source-sink effect, where the population at the central site attains a value near the carrying capacity, while the populations at neighboring sites drop exponentially with jmj. For greater values of p, if a is smaller than the critical value a = 0.36 the population at all sites attains a value near the carrying capacity. This behavior corresponds to the rescue effect, where dispersal from a certain site leads to complete colonization of neighboring empty sites. For a > 0.36, the metapopulation is lead to extinction by increasing the dispersal probability, see Fig. 4. We will study the behavior of the system (4) with respect to the parameters b and p, for the two values a = 0.3 and a = 0.6, which correspond to a small and a large value of the Allee threshold of the prey, as mentioned in the previous section. The bifurcation diagram for a = 0.3, obtained by starting with the initial conditions defined above, is given in Fig. 5. In this figure the domains of different behavior of the system, for pairs of values b, p are shown. For values of the parameters in the domain SSN, the system approaches to a stable node, which corresponds to a localized solution where the population of the prey at the cental site almost equals the carrying capacity and the other sites possess a population smaller than the Allee threshold. This behavior corresponds to a source-sink effect. Domain SSF corresponds to a source-sink effect too, but in this case the asymptotic equilibrium is a stable focus. For b < 0.666 the transition from a node to focus occurs smoothly, while for b > 0.666 the asymptotic behavior jumps from the stable node initially (for p = 0) connected to the point (1, 1) to a stable focus connected to the point (x*, y*), see Fig. 6. For values of the parameters in the domain IC, the system tends asymptotically to an invariant circle which corresponds to the invariant circle of the case p = 0 of Section 4, supporting quasiperiodic orbits where all populations oscil-
Fig. 4. The bifurcation diagram for b = 0 and y = 0.
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Fig. 5. The bifurcation diagram for a = 0.3.
Fig. 6. The y-value at the stable fixed point for (a) b = 0.5 and (b) b = 0.67.
late, see Fig. 7. During these oscillations, however, the population of the prey at the non-central sites remains very small. Domain E corresponds to extinction of the populations at all sites. This happens when either the dispersal probability p or the predation parameter b are large. In Fig. 8, the bifurcation diagram for a = 0.6 and the same initial conditions is shown. The domain SSN, for small values of p and b corresponds to a source-sink effect leading to a stable node, while domain E corresponds to extinction. This means that, in this case where the Allee effect operates strongly, survival of the metapopulation is possible only for very small dispersal and/or predation. In all cases the transitions from one domain to the other happen abruptly, by means of bifurcations of the fixed points. Comparing the behavior of the system in presence of predation to the one for b = 0 and y = 0, i.e. Fig. 4, we see that, for small values of p we have localization of the prey population in the central site, corresponding to various stable dynamical configurations, i.e. nodes, foci or limit cycles but with increasing p no rescue effect is observed due to predation.
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Fig. 7. The projection of the invariant circle in the x(0), x(1), y subspace.
Fig. 8. The bifurcation diagram for a = 0.6.
6. Conclusions In the present paper we construct a mapping model for a system including a specialist predator and its prey. The predator dynamics is of logistic type, while the prey is subjected to the Allee effect. For small values of prey population, this effect manifests itself strongly and both populations are led to extinction. If the Allee threshold is small and the initial population of the prey is rather large, there are several stable dynamical states. For small values of the predation parameter b, the point (1, 1), which corresponds to the carrying capacity of the environment, is stable. For larger values of b, a second stable fixed point appears for smaller values of both populations. For 0.7 < b < 0.789 the populations
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exhibit stable quasiperiodic oscillations, whose amplitude increases with b. These large amplitude oscillations implement the paradox of enrichment, as commented in Section 4. For still larger values of b, both populations extinct. When the value of the minimal viable prey population is large, the Allee effect dominates. In this case, besides to (0, 0) which corresponds to the extinction of both populations, for small b the point (1, 1) is the only non-trivial stable fixed point, but its basin of attraction is significantly smaller and is restricted in the region of large values of the prey population. If, however, the predation parameter is large, i.e. b > 0.4, all initial conditions lead to extinction of both populations. Finally, for non-zero probability of dispersal, even for small Allee threshold, predation is an obstacle to the colonization of neighboring sites by the prey, i.e. to the rescue effect.
Acknowledgements We thank Drs. S. Sgardelis, J. Halley, E. Meletlidou and V. Koukouloyannis for their valuable help. This work is partially supported by ‘‘Pythagoras’’ Research Programme Nr. 21878 of the Greek Ministry of Education and the European Union.
References [1] Allee WC. Animal aggregations, a study in general sociology. Chicago: University of Chicago Press; 1931. [2] Allee WC. The social life of animals. London: Heinemann; 1938. [3] Fowler CW, Baker JD. A review of animal populations dynamics at extremely reduced population levels. Rep Int Whal Comm 1991;41:545–54. [4] Stephens PA, Sutherland WJ, Freckleton RP. What is the Allee effect? Oikos 1999;87:185–90. [5] Taylor CM, Hastings A. Allee effect in biological invasions. Ecol Lett 2005;8:895–908. [6] Colato A, Mizrahi SS. Effects of random migration in population dynamics. Phys Rev E 2001;64:011901. [7] Lo´pez-Ruiz R, Fournier-Prunaret D. Indirect Allee effect, bistability and chaotic oscillations in a predator–prey discrete model of logistic type, 2004, Archive preprint nlin.CD/0406019. [8] Hadjiavgousti D, Ichtiaroglou S. Existence of stable localized structures in population dynamics. Chaos, Solitons & Fractals 2004;21:119–31. [9] Hadjiavgousti D, Ichtiaroglou S. Allee effect in population dynamics: Existence of breather-like behavior and control of chaos through dispersal, Int J Bifurcation Chaos 2006; 16(7), in press. [10] Whitehead DR, Duffield RM. An unusual specialized predator prey association (Coleoptera: Coccinellidae, Chrysomelidae): Failure of a chemical defense and possible practical application. Coleopt Bull 1982;36:96–7. [11] Steghaus-Kovac S, Maschwitz U. Predation on earwigs: A novel diet specialization within the genus Leptogenys (Formicidae: Ponerinae. Ins Soc 1993;40:337–40. [12] Ko¨pf A, Rank NE, Roininen H, Tahvanainen J. Defensive larval secretions of leaf beetles attract a specialist predator Parasyrphus nigritarsis. Ecol Entomol 1997;22:176–83. [13] Hieber CS, Wilcox RS, Boyle J, Uetz GW. The spider and fly revisited: ploy–counterploy behavior in a unique prey–predator system. Behav Ecol Sociobiol 2002;53:51–60. [14] Rosenzweig ML. Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time. Science 1971;171:385–7. [15] May RM. Limit cycles in predator–prey communities. Science 1972;177:900–2. [16] Gilpin ME, Rosenzweig ML. Enriched predator–prey systems: theoretical stability. Science 1972;177:902–4. [17] Abrams PA, Walters CJ. Invulnerable prey and the paradox of enrichment. Ecology 1996;77:1125–33. [18] Nisbet RM, de Roos AM, Wilson WG, Snyder RE. Discrete consumers, small scale resource heterogeneity, and population stability. Ecol Lett 1998;1:34–7. [19] Wolkowicz GSK. Bifurcation analysis of a predator–prey system involving group defence. SIAM J Appl Math 1988;48:592–606. [20] Huang J, Xiao D. Analyses of bifurcations and stability in a predator–prey system with Holling type–IV functional response. Acta Math Appl Sinica 2004;20:167–78. [21] Kent A, Doncaster CP, Sluckin T. Consequences for predators of rescue and Allee effects on prey. Ecol Model 2003;162:233–45. [22] Owen MR, Lewis MA. How predation can slow, stop or reverse a prey invasion. Bull Math Biol 2000;1:1–35. [23] Fagan WF, Lewis MA, Neubert MG, van den Driessche P. Invasion theory and biological control. Ecol Lett 2002;5:148–57. [24] de Roos AM, McCauley E, Wilson WG. Pattern formation of the spatial scale of interaction between predators and their prey. Theor Pop Biol 1998;53:108–30. [25] Petrovskii SV, Morozov AY, Venturino E. Allee effect makes possible patchy invasion in a predator–prey system. Ecol Lett 2002;5:345–52.
Chaos, Solitons and Fractals 36 (2008) 334–342 www.elsevier.com/locate/chaos
Allee effect in a prey–predator system Despina Hadjiavgousti, Simos Ichtiaroglou
*
Department of Physics, University of Thessaloniki, 54124, Greece Accepted 19 June 2006
Abstract We construct a model of a specialist predator and its prey species for the case of non-overlapping generations and we assume that the prey obeys to the Allee effect. We observe that, for small values of the minimal viable population of the prey, there are stable dynamical configurations corresponding either to stable fixed points or to a stable limit cycle which supports quasiperiodic orbits. In this last case we observe the paradox of enrichment, where an initial increase of the population of the prey may lead to cycles of very fast increase and subsequent decrease of the predator population. This behavior increases the probability of extinction due to unpredictable external factors. We consider also non-zero dispersal probability of the prey to neighboring sites and observe that predation causes the disappearance of a rescue effect, even when the Allee threshold is small. 2006 Elsevier Ltd. All rights reserved.
1. Introduction The reduction of the per capita growth rate of a population of a biological species at densities smaller than a critical value is known as the Allee effect, see e.g. [1–5]. The main cause of the Allee effect is the difficulty in finding mates between the individuals of a species at low population densities. Other causes may be reduced defense against predators, special trends of social behavior, etc. The dynamical description by difference equations (maps) is valid in the case of populations with distinct, non-overlapping generations, for example some insect populations. Such a dynamical description has often been used in population dynamics in connection to logistic type maps, e.g. [6,7]. In two previous papers [8,9] a number of identical sites, appropriate for colonization by a biological species obeying the Allee effect, was considered. It was assumed that initially the central site carries a population equal to the carrying capacity of the environment, while all other sites are empty. For non-zero dispersal probability three final configurations were observed: (a) the source-sink effect, where almost all individuals remain at the central site, while the populations at the nearby sites attain very small values, significantly smaller than the Allee threshold, (b) the rescue effect, where complete colonization at all sites occurs through dispersal from the central one and (c) the extinction of the metapopulation for relatively large values of the dispersal probability. In [8] the Allee effect was modeled by a third degree polynomial map, while a polynomial of fifth degree was used in [9]. In the latter case, there was a control of chaotic behavior by increasing the dispersal probability.
*
Corresponding author. E-mail address: [email protected] (S. Ichtiaroglou).
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.053
D. Hadjiavgousti, S. Ichtiaroglou / Chaos, Solitons and Fractals 36 (2008) 334–342
335
In the present paper, in order to study the combined influence of predation and the Allee effect on a population, we construct a prey–predator model, compatible with the following assumptions: (a) both populations have non-overlapping generations, so the dynamics is described by a discrete map, (b) the dynamics of the prey, in absence of predator, is subjected to the Allee effect, (c) the predator is specialist, see e.g. [10–13] and its dynamics is of logistic type, (d) the normalized value (1, 1) is a fixed point of the map and (e) the square U = [0, 1] · [0, 1] remains invariant under the map. We study the fixed points and their stability and obtain phase portraits for various values of the parameters. We observe that, for small values of the Allee threshold, i.e. the minimal viable population of the prey, there are stable dynamical configurations corresponding either to stable fixed points or to a limit cycle which supports quasiperiodic orbits. This latter behavior relates to the paradox of enrichment [14–20], where an initial increase of the prey population leads to cycles of very fast increase and subsequent decrease (‘‘boom and bust’’, [21]) of the predator population, and this fact increases the probability of extinction of the predator due to unpredictable external factors. In contrast, for relatively large values of the Allee threshold or the predation parameter, both populations are led to extinction. Finally, we consider a chain of five identical sites, suitable for colonization by the prey species and non-zero probability of dispersal from the central site to the neighboring, initially empty sites. We observe that, even for small values of the minimal viable population of the prey, predation prohibits the appearance of a rescue effect, in accordance to other models, where predation may slow, stop or even reverse the invasion of a prey [22,23] or create a patchy distribution of its population [24,25].
2. The model We introduce a mapping model for prey–predator populations localized at one site. We consider the map xnþ1 ¼ F 1 ðxn ; y n Þ ¼ xn xn ð1 xn Þða xn Þ bxn y n ð1 xn Þ; xn y n ð3 y n Þ ; y nþ1 ¼ F 2 ðxn ; y n Þ ¼ 1 þ xn
ð1Þ ð2Þ
where x, y 2 U = [0, 1] · [0, 1], Fi : U ! U, i = 1, 2, i.e. the unit square U remains invariant. The variables xn, yn represent, respectively, the normalized populations of a prey and its predator at the nth generation. The functions F1, F2 indicate the relation between the reproduction and death rates of the two populations and the predation rate at a certain localized site, appropriate for colonization. The term xng1(xn) = xn[1 (1 xn)(a xn)] stands for the rate of the increase of the prey population in absence of predator and indicates that the prey obeys to the Allee effect. If it is smaller than a critical value x = a < 1, it will extinct. The parameter a is the Allee threshold and equals to the minimal viable population of the prey. The term xng2(xn, yn) = yn[bxn(1 xn)] represents the rate of decrease due to predation. This rate is proportional to the density yn and the predation ability g2 of the predator. This ability ascribes the relative predation rate, i.e. the number of individuals killed by the predator at one iteration of the map. The parameter b is the predation parameter. In this paper we choose 0.1 < b < 0.8. The dynamics of the prey in absence of predation (b = 0) has been studied in detail in Ref. [8]. The term yng3(xn, yn) = ynxn(3 yn)/(1 + xn) stands for the variation of the predator density with respect to the prey population. Note that, if the prey disappears, the population of the predator extincts at one generation. The predator is specialist, i.e. it feeds on this particular prey only, so that the point (0, 0) is an asymptotically stable fixed point and corresponds to the extinction of both populations. In addition, we assume the dynamics of the predator being of the logistic type and (1, 1) is also a fixed point of the map.
3. Fixed points and stability The fixed points of the map (2) are determined as the solutions of the system x = F1(x, y), y = F2(x, y) which belong in U. Let ! oF 1 oF 1 DF ¼
ox
oy
oF 2 ox
oF 2 oy
be the matrix of the linearized map at a fixed point. This point is a stable (unstable) node if both eigenvalues of DF are real and absolutely smaller (greater) than 1. If the eigenvalues are real and one is smaller while the other is greater than 1, it is a saddle point. If the eigenvalues are complex conjugates with real part smaller (greater) than 1, the point is a stable (unstable) focus. The map (2) has the following fixed points:
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• (x, y) = (0, 0): The eigenvalues are 1 a, 0, so the point is a stable node. • (x, y) = (1, 0): The eigenvalues are a, 3/2 and the point is a saddle. The axis y = 0 and the line x = 1 are the stable and unstable manifolds, respectively. • (x, y) = (1, 1): The eigenvalues are a + b, 1/2, so the point is a stable node for a + b < 1. For a + b > 1 it is a saddle and the lines x = 1 and x = y are the stable and unstable manifold, respectively. • (x, y) = (a, 0): The eigenvalues are 1 + a(1 a), 3a/(1 + a). The point is a saddle for a < 0.5 with the axis y = 0 the unstable manifold. The stable manifold depends on b. For a > 0.5 it is an unstable node. • (x, y) = (x*, y*) where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ a þ by ; y ¼ ½ð2b aÞ ð2b aÞ2 þ 4bð1 2aÞ=2b: These points are taken into account when they are real and belong to U. Their stability will be studied through the phases portraits of the map.
4. Dynamics of the map We will study numerically the map (2) for two values of a, one smaller and one greater than the bifurcation value 0.5, and for several values of b. These cases correspond to a small and a large value of the minimal viable population of the prey, i.e. the Allee threshold, respectively. In the second case the Allee effect is expected to manifest itself more strongly. We select the values a = 0.3 and a = 0.6. In all figures the stable nodes are represented by circles, the unstable node by a square, and the saddles by rhombuses. In Fig. 1, the phase portraits of the map (2) for a = 0.3 are shown. For this value of a, the fixed point at (a, 0) is a saddle for all values of b and the state space is partitioned in two domains by the separatrix which coincides with the stable manifold of this point. All the initial conditions lying in the domain at the left side, i.e. corresponding to small initial values of the prey population, lead to the stable node at (0, 0), i.e. to extinction of both populations. This domain is the basin of attraction of (0, 0), where the influence of the Allee effect for small x is evident. For b < 0.666, the domain at the right of the separatrix is the basin of attraction of the stable node at (1, 1) (Fig. 1a and b). Thus, for small values of the predation parameter, the populations are lead either to extinction or to the carrying capacity of the environment, depending on whether the prey population is initially small or not. We observe also that, as the predation parameter increases, the separatrix bends, resulting to an increase of the extinction domain. For b = 0.666 a saddle-focus bifurcation occurs and the two fixed points (x*, y*) appear, as can be seen at Fig. 1c for b = 0.68. The final state of the system, besides extinction, may be either the point (1, 1) or the stable focus. For b = 0.7 (Fig. 1d) we have a + b = 1 and a subcritical bifurcation occurs. The saddle point generated by the saddle-focus bifurcation collides with the stable node at (1, 1) interchanging stability. The point (1, 1) becomes a saddle and the stable node moves out of the square U. At the same time, the stable focus undergoes a Hopf bifurcation becoming unstable and a stable invariant limit cycle, supporting quasiperiodic orbits, appears (Fig. 1e). Increasing b further, the invariant circle increases in size. Thus, it is possible for the system to perform stable quasiperiodic oscillations, during which, while the prey population remains sufficiently large and the Allee effect is not operating, the predator population attains extremely small values. Although, in this model the predator does not extinct during the oscillations, in a realistic situation these small values may put a threat of extinction due to unpredictable external factors. This behavior corresponds to the paradox of enrichment, where an initially increased prey population leads to cycles of very fast increase and subsequent decrease of the predator population. As mentioned in the introduction, this fact increases the probability of extinction of the predator due to unpredictable external factors. This behavior is maintained up to b ’ 0.798, where the invariant circle touches the stable manifolds of the saddles at (a, 0) and (1, 1) and disappears. At the same time, a heteroclinic connection is formed between these two saddle points. For still larger values of b, the stable manifold of (a, 0) winds around the unstable focus, while the unstable manifold of (1, 1) is connected to the stable node at (0, 0). This shift of the invariant manifolds by varying b is shown in Fig. 2. For values of the predation parameter b > 0.798, the node at (0, 0) is the only attracting fixed point, so all initial conditions in the unit square U lead to extinction, with the exception of the points on the stable manifold of (a, 0), which are of measure zero. As is apparent from the spiraling orbits in Fig. 1f, both populations may attain significantly large values, before ultimate extinction. In Fig. 3, the phase portraits of system (2) for a = 0.6 are shown. For a > 0.5 and a + b < 1, there is always only one fixed point (x*, y*) 2 U, which is a saddle. So, in Fig. 3a for b = 0.3, the stable configurations correspond to the points
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Fig. 1. The state space for a = 0.3 and b = 0.1 (a), b = 0.6 (b), b = 0.68 (c), b = 0.7 (d), b = 0.72 (e), b = 0.8 (f).
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Fig. 2. The asymptotic manifolds for (a) b = 0.72, (b) b = 0.798 (the heteroclinic connection) and (c) b = 0.85.
Fig. 3. The state space for a = 0.6 and b = 0.3 (a) and b = 0.5 (b).
(0, 0) and (1, 1). Most of the initial conditions lead to extinction, but a set of initial conditions for large values of the prey population, leads to the normalized value (1, 1) which is the carrying capacity of the environment. Comparing this behavior to the one shown in Fig. 1a and b, we observe that, if the Allee effect manifests itself strongly (large a), a large population of the prey is demanded for survival of both populations. For a + b = 1, a subcritical bifurcation occurs and the saddle (x*, y*) collides with the stable node at (1, 1) and interchange their stability. Thus, for values of b > 0.4 the only stable configuration in U is the node at (0, 0) and all initial conditions lead to extinction, as is shown in Fig. 3b for b = 0.5. In this case too, for large values of prey, the population of the predator may attain significant values, before it extincts.
5. Patchy distribution of the prey In this section, we consider a chain consisting of five identical sites, which are appropriate for colonization by the prey species, while the predator explores all of them and predates on the whole metapopulation of the prey at all sites with equal probability. We define each site by an integer m, such that 2 6 m 6 2 and, at time n, the prey population at this site is xnðmÞ . Each individual of the prey either emigrates to the nearest neighbor sites to the right or left, with equal probability p, or remains at the same site, with probability 1 2p. The probability of dispersal to a non-nearest neighboring site equals to zero. An individual at the boundary sites of the chain m = ±2, may either move to the site m = ±1, or remain at the same site, or escape from the population. The mapping for this system is described by the following equations:
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xnþ1 ¼ pðxðmþ1Þ þ xðm1Þ Þ þ ð1 2pÞF 1 ðxðmÞ n n n ; y n Þ; 2 P
y n ð3 y n Þ 1þ
2 P
ð3Þ
xnðmÞ
m¼2
y nþ1 ¼
339
ð4Þ
ðmÞ
xn
m¼2
where ðmÞ ðmÞ ðmÞ ðmÞ ðmÞ ðmÞ F 1 ðxðmÞ n ; y n Þ ¼ xn xn ð1 xn Þða xn Þ bxn y n ð1 xn Þ:
Initially, at n = 0, we consider that the population of the prey at the central site m = 0 equals the carrying capacity of ð0Þ ðmÞ the site in absence of the predator, x0 ¼ 1, while all other sites are empty, x0 ¼ 0, m 5 0. The initial value of the predator population is also considered to be y0 = 1. In Ref. [8] we studied the behavior of the above model in absence of predation, i.e. for b = 0 and y = 0 for the above initial conditions. It was shown that, for sufficiently small values of p there is a localization of the metapopulation at the central site, which corresponds to the source-sink effect, where the population at the central site attains a value near the carrying capacity, while the populations at neighboring sites drop exponentially with jmj. For greater values of p, if a is smaller than the critical value a = 0.36 the population at all sites attains a value near the carrying capacity. This behavior corresponds to the rescue effect, where dispersal from a certain site leads to complete colonization of neighboring empty sites. For a > 0.36, the metapopulation is lead to extinction by increasing the dispersal probability, see Fig. 4. We will study the behavior of the system (4) with respect to the parameters b and p, for the two values a = 0.3 and a = 0.6, which correspond to a small and a large value of the Allee threshold of the prey, as mentioned in the previous section. The bifurcation diagram for a = 0.3, obtained by starting with the initial conditions defined above, is given in Fig. 5. In this figure the domains of different behavior of the system, for pairs of values b, p are shown. For values of the parameters in the domain SSN, the system approaches to a stable node, which corresponds to a localized solution where the population of the prey at the cental site almost equals the carrying capacity and the other sites possess a population smaller than the Allee threshold. This behavior corresponds to a source-sink effect. Domain SSF corresponds to a source-sink effect too, but in this case the asymptotic equilibrium is a stable focus. For b < 0.666 the transition from a node to focus occurs smoothly, while for b > 0.666 the asymptotic behavior jumps from the stable node initially (for p = 0) connected to the point (1, 1) to a stable focus connected to the point (x*, y*), see Fig. 6. For values of the parameters in the domain IC, the system tends asymptotically to an invariant circle which corresponds to the invariant circle of the case p = 0 of Section 4, supporting quasiperiodic orbits where all populations oscil-
Fig. 4. The bifurcation diagram for b = 0 and y = 0.
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Fig. 5. The bifurcation diagram for a = 0.3.
Fig. 6. The y-value at the stable fixed point for (a) b = 0.5 and (b) b = 0.67.
late, see Fig. 7. During these oscillations, however, the population of the prey at the non-central sites remains very small. Domain E corresponds to extinction of the populations at all sites. This happens when either the dispersal probability p or the predation parameter b are large. In Fig. 8, the bifurcation diagram for a = 0.6 and the same initial conditions is shown. The domain SSN, for small values of p and b corresponds to a source-sink effect leading to a stable node, while domain E corresponds to extinction. This means that, in this case where the Allee effect operates strongly, survival of the metapopulation is possible only for very small dispersal and/or predation. In all cases the transitions from one domain to the other happen abruptly, by means of bifurcations of the fixed points. Comparing the behavior of the system in presence of predation to the one for b = 0 and y = 0, i.e. Fig. 4, we see that, for small values of p we have localization of the prey population in the central site, corresponding to various stable dynamical configurations, i.e. nodes, foci or limit cycles but with increasing p no rescue effect is observed due to predation.
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Fig. 7. The projection of the invariant circle in the x(0), x(1), y subspace.
Fig. 8. The bifurcation diagram for a = 0.6.
6. Conclusions In the present paper we construct a mapping model for a system including a specialist predator and its prey. The predator dynamics is of logistic type, while the prey is subjected to the Allee effect. For small values of prey population, this effect manifests itself strongly and both populations are led to extinction. If the Allee threshold is small and the initial population of the prey is rather large, there are several stable dynamical states. For small values of the predation parameter b, the point (1, 1), which corresponds to the carrying capacity of the environment, is stable. For larger values of b, a second stable fixed point appears for smaller values of both populations. For 0.7 < b < 0.789 the populations
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exhibit stable quasiperiodic oscillations, whose amplitude increases with b. These large amplitude oscillations implement the paradox of enrichment, as commented in Section 4. For still larger values of b, both populations extinct. When the value of the minimal viable prey population is large, the Allee effect dominates. In this case, besides to (0, 0) which corresponds to the extinction of both populations, for small b the point (1, 1) is the only non-trivial stable fixed point, but its basin of attraction is significantly smaller and is restricted in the region of large values of the prey population. If, however, the predation parameter is large, i.e. b > 0.4, all initial conditions lead to extinction of both populations. Finally, for non-zero probability of dispersal, even for small Allee threshold, predation is an obstacle to the colonization of neighboring sites by the prey, i.e. to the rescue effect.
Acknowledgements We thank Drs. S. Sgardelis, J. Halley, E. Meletlidou and V. Koukouloyannis for their valuable help. This work is partially supported by ‘‘Pythagoras’’ Research Programme Nr. 21878 of the Greek Ministry of Education and the European Union.
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