stochastic predator–prey model with allee effect on prey

Nonlinear Analysis: Real World Applications 14 (2013) 768–779

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Nonlinear Analysis: Real World Applications

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Stochastic predator–prey model with Allee effect on preyPablo Aguirre a,∗, Eduardo González-Olivares b, Soledad Torres c

a Departmento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chileb Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chilec CIMFAV Facultad de Ingeniería, Universidad de Valparaíso, Casilla 123-V, Valparaíso, Chile

a r t i c l e i n f o

Article history:Received 15 August 2011Accepted 31 July 2012

Keywords:Stochastic predator–prey modelAllee effectParameter estimation

a b s t r a c t

We study a predator–prey model with the Allee effect on prey and whose dynamics isdescribed by a system of stochastic differential equations assuming that environmentalrandomness is represented by noise terms affecting each population. More specifically,we consider a term that expresses the variability of the growth rate of both species dueto external, unpredictable events. We assume that the intensities of these perturbationsare proportional to the population size of each species. With this approach, we prove thatthe solutions of the system have sample pathwise uniqueness and bounded moments.Moreover, using an Euler–Maruyama-type numerical method we obtain approximatedsolutions of the system with different intensities for the random noise and parameters ofthe model. In the presence of a weak Allee effect, we show that long-term survival of bothpopulations can occur. On the other hand, when a strong Allee effect is considered, weshow that the random perturbations may induce the non-trivial attracting-type invariantobjects to disappear, leading to the extinction of both species. Furthermore, we also findthe Maximum Likelihood estimators for the parameters involved in the model.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In population dynamics, the Allee effect (named after the seminal works of the zoologist and ecologist Warder ClydeAllee [1]) refers to a process that reduces the growth rate for small population densities [2]. Different mechanisms cangenerate Allee effects, such as the difficulty of finding mates, social interaction or predation [3], to name a few examples;see also Table 1 in [4]. These mechanisms have been mainly studied as singular entities and usually describe a situation inwhich the population growth rate decreases under someminimum critical density [5], or when a limited population growthcapacity is observed [6]. In some cases, an Allee effectmay even cause a population growth rate to become negative, creatingan extinction threshold that the population has to overcome in order to survive [5,7,8]. This critical situation is also knownas a strong Allee effect [4] or critical depensation [7]. In this case, a species may have a larger tendency to be less able toovercome these additional mortality causes or to have a slower recovery, and to be more prone to extinction than otherspecies [2]. On the other hand, the weak Allee effect (also known as noncritical depensation [7]) does not have a thresholdpopulation level, although there is still a reduced (but positive) growth rate [2]. Notice that the Allee effect may also beknown as a negative competition effect [9]; in fisheries sciences, it is called depensation [7,8] and in epidemiology, its analogyis the eradication threshold, the population level of susceptible individuals below which an infectious illness is eliminatedfrom a population [5].

∗ Corresponding author.E-mail addresses: [email protected], [email protected] (P. Aguirre), [email protected] (E. González-Olivares), [email protected]

(S. Torres).

1468-1218/$ – see front matter© 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2012.07.032

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P. Aguirre et al. / Nonlinear Analysis: Real World Applications 14 (2013) 768–779 769

Typically, the role of the Allee effect on the dynamics of populations has been extensively studied by means of deter-ministic models [10,5,11–13]. However, it is known that the Allee effect can also interact with environmental stochasticity[14–17]. Random environmental conditions, such as catastrophic events [18] or alien species invasions [15], can increase theamplitude of population fluctuations and even drive a population to extinction [19,8,20]. Nevertheless, in spite of the grow-ing interest from the Population Dynamics community on the subject [21,3,13], very few results are available for stochasticmodels with both predation and the Allee effect [22].

In this paper we propose a stochastic extension of the predator–prey model first studied in [23,10]. More concretely,we analyze the effect produced by a random noise associated to the environment of the populations, acting on the growthrates of preys and predators, respectively. Due to the non-linearities of the model, we implement an Euler–Maruyama-typenumerical method [24] and simulate the dynamics of the stochastic model for different intensities of the random noiseand parameters of the model. In this way, we are able to gain insight into the complicated effects and consequences of thestochastic noise on the behavior of both populations.

Moreover, we derive a sequence of explicit maximum likelihood estimators for the parameters of the model based on aregular Euler approximation [24]. Thismethod has beenwidely used in stochastic biologicalmodels. For instance, in [25], theauthor considers themaximum likelihood estimation for a simple stochasticmodel for the spread of an epidemic in a closed,homogeneously mixing population. In [26] a continuous-time model is constructed to study the spread of an infectiousdisease. The inference procedure makes use of martingale estimating equations, and explicit expressions for the estimatesare obtained. On the other hand, in [27] the author introduces a Cellular Automata (CA) to simulate the transition rulesof stochastic parameters for the first-order linear kinetics and the Lotka–Volterra equations that describe prey–predatorinteractions. The simulation results from the CA models are shown to be in close agreement with those of the continuum-based models. A generalization of Lotka–Volterra models by using stochastic nonlinear differential equations was studiedin [28]. They studied two andmultidimensional systems of stochastic differential equations, which can be used in statisticalinference.

This paper is structured as follows. The model is presented in Section 2. In Section 3 we summarize the main results forthe original deterministic model. In Section 4we present some preliminary results of the stochastic model such as existenceand uniqueness of solutions. The numerical simulations for the model are given in Section 5, performed by the Split-StepBackward Euler method [29]; furthermore an interpretation of the results is given. Finally, in Section 6, the MaximumLikelihood estimation for the parameters of the model is found.

2. The model

There exist different mathematical ways to formalize the Allee effect in deterministic continuous time models.One of these methods is just by adding a suitable term −A(x) in the logistic equation [8], obtaining the expressionx = r x (1 − x/k) − A(x) x. This approach leads to what is commonly known as an additive Allee effect [23,10,2]. In our case,the function A(x) takes the form

A(x) =m

x + b,

where r is the intrinsic growth rate or biotic potential of the population x, k is the environment carrying capacity of x [30], andm and b are constants that indicate the severity of the Allee effect as deduced in [31,2]. Hence, let us consider the followingdeterministic predator–prey model described by the system of ordinary differential equations

dxdt

= P(x, y) :=

r1 −

xk

−

mx + b

x −

qxyx2 + a

,

dydt

= Q (x, y) := sy1 −

ynx

;

(1)

where (x(t), y(t)) ∈ A = {(x, y)| x > 0, y ≥ 0} denote prey and predator densities, respectively, as functions of time t , andµ = (r, a, b, k,m, n, q, s) ∈ R8

+are parameters. The vector field F(x, y) := (P(x, y),Q (x, y))t induces a flow ϕt on A that

determines the dynamics.Let us describe the ingredients inmodel (1) inmore detail. In the first place, it is assumed that the preymay create a group

defense mechanism to counteract an Allee effect generated by predation [32]. This phenomenon is a type of antipredatorbehavior (APB) characterized by a decrease in the predators population due to the increased ability of the prey to betterdefend or disguise themselves when their number is large enough [33–36]. Notice that the Allee effect and APBmechanismsare quite compatible. Indeed, the evidence strongly supports the notion of Allee effects induced by predation, see forinstance [37–40] and Table 2 in [32] for a wide range of predator-driven Allee effects. In such case, the predator rateconsumption is conveniently modeled by a non-monotonic functional response; see [23,10] and the references therein forfurther discussion. In this work we use the function

h(x) =qx

x2 + a,

770 P. Aguirre et al. / Nonlinear Analysis: Real World Applications 14 (2013) 768–779

also employed in [41,34–36] and corresponding to a Holling type IV functional response [42]. In particular, q is the predatormaximum consumption rate per capita, i.e., the maximum number of prey that can be eaten by a predator in each time unit,and a is the number of prey necessary to achieve one-half of the maximum rate q.

The last aspect in our Eqs. (1) is a feature of Leslie type predator–prey models [43] or the Leslie–Gower model [44]. Inaddition to the intrinsic growth rate of the predator s, one considers that the conventional environmental carrying capacityKy of the predator is proportional to prey abundance x [30], that is, Ky = nx, as in the May–Holling–Tanner model [45] andother models recently analyzed [11,41,46].

Nevertheless, the aim of this paper is to analyze the effect produced by a random noise associated to the environment ofthe populations, acting on the growth rates of preys and predators, respectively. Wemodel these perturbations bymeans ofindependent Gaussianwhite noises. This means to extend the model (1) and consider a system of Itô’s stochastic differentialequations in the form:

dx =

r1 −

xk

x −

mxx + b

−qxy

x2 + a

dt + σ1x dW1(t);

dy =

s1 −

ynx

ydt + σ2y dW2(t);

(2)

where W (t) = (W1(t),W2(t)| t ≥ 0) is a two-dimensional Brownian motion and parameters (σ1, σ2) ∈ R2+

representthe intensity of the perturbation. This multiplicative-type random noise in (2) has already been used in previous works forLotka–Volterra type models in [28,47–50], and recently generalized to the form g(x) = σ xθ , θ ∈ [0, 0.5] in [51], and to thenon-autonomous case g(x, t) = σ(t) x(t) in [52], for one-dimensional logistic models though.

In the following sections, it will be more convenient to express the system (2) in the following equivalent vector form:dX(t) = F (x(t), y(t)) dt + G(x(t), y(t)) dW (t), (3)

where X(t) =

x(t)y(t)

,G(x(t), y(t)) =

σ1 x(t) 0

0 σ2 y(t)

, andW (t) =

W1(t)W2(t)

is a two-dimensional Brownian motion.

An Euler–Maruyama type scheme is implemented in Section 5 to study numerically the dynamics given by (3). Fur-thermore, in Section 6, we find the maximum likelihood estimator θ =

r, a, b, k, m, n, q, s, σ 2

1 , σ 22

Nfor the vector of

parameters θ =r, a, b, k,m, n, q, s, σ 2

1 , σ 22

in the discrete model given by the Euler approximation with N observations.

The equations for θ are explicit, which allows us to implement the resolution of this system, yielding a practical method forestimating θ . We also present numerical results based on simulated data which show the efficiency of the method.

3. Main results for the deterministic model

In what follows we refer to [23,10] for more details.It is clear that P andQ areC∞ functions inA, therefore the vector field F satisfies a local Lipschitz-type condition in every

point of A. Moreover, notice from (1) and (3) that the axis y = 0 is invariant under the dynamical system associated to F . Inparticular, this is also true for the stochastic system (2). On the other hand, in [23] it is proven that F has a C∞-equivalentpolynomial extension (see [53,54]) in A = {(x, y)| x ≥ 0, y ≥ 0}. In other words, there is a polynomial vector field that iswell defined in the axis x = 0 and is qualitatively equivalent to F in A. Furthermore, the extended vector field is invariantin the axis x = 0 and is also bounded in A, in the sense that no trajectory diverges to infinity. It follows that every orbit ofF in the interior of the first quadrant remains forever inside that region.

It is also shown that b, k and L =br−m

s are the most significant parameters of the predator–prey system (1). In particular,the sign of L determines the type of Allee effect on the prey population in the absence of predators.

3.1. Weak Allee effect

For L > 0, the origin is repulsing in the x-axis, hence there is a weak Allee effect on the prey population. However, thisweak Allee effect does not induce extinction in the two-dimensional system (1). On the contrary, the slow growth rate ofthe prey for low densities also affects negatively in the predators success for finding their prey. This allows a slow (butsustained) recovery of the preys (and ultimately, also of the predators). From that point of view, this case can be thoughtof as an extension of a weak Allee effect (initially) in the absence of predation to an ecologically sustainable predator–preysystem. This is illustrated in Fig. 1 where the origin has a separatrix dividing a repelling parabolic sector and a hyperbolicsector [53]; see Lemma 1, part iii, in [23] for details. This means that every solution with initial conditions (x0, y0) in theinterior of the first quadrant near the origin ultimately tends towards the only attractor p ≈ (0.173202, 0.173202), allowingboth populations to survive.

3.2. Strong Allee effect

The overall dynamics in the phase plane for a strong Allee effect is illustrated in Fig. 2. This scenario occurs when L < 0.Under certain conditions, there are three (consecutive) equilibria p0, p−, p+ in the x-axis (not labeled in Fig. 2): p0 = (0, 0)is a non-hyperbolic attractor, that is, a locally asymptotically stable equilibrium with zero eigenvalues; p− is a repelling


Fig. 1. Numerical simulations of system (1) with weak Allee effect for a = c = k = m = n = q = s = 1, b = 0.5 and r = 2. Every solution in the interiorof the first quadrant converges to a unique equilibrium denoted by p.

Fig. 2. Numerical simulations of system (1) with strong Allee effect where a = 3, b = n = s = 1, k = 6.79211,m = 4.07697, q = 4.05116, r = 4.02708.In this scenario, two infinitesimal limit cycles surround the equilibrium f1 and a third periodic orbit lies between the blue and green trajectories andsurrounds f2 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

node and p+ is a saddle. Hence, a strong Allee effect is present on the prey population, if y = 0, with p− being the criticalAllee threshold.

However, in the presence of predators, i.e., for y > 0, there are another four equilibria in the diagonal y = x, namely,two saddles s1 = (s1, s1) and s2 = (s2, s2), and two foci f1 = (f1, f1) and f2 = (f2, f2), with 0 < s1 < f1 < s2 < f2.Moreover, it is known that the focus f1 can undergo a generalized Hopf bifurcation [54]. This produces a stable limit cycleΓ1 that surrounds both the stable focus f1 and an unstable limit cycle Γ2 which serves as the separatrix between their basinsof attraction; i.e., there is a range for population sizes for which there exists autoregulation for the predator–prey system.Note that the limit cycles Γ1 and Γ2 are infinitesimal and, therefore, hard to visualize in the scale of Fig. 2.

A third (unstable) limit cycle Γ3 surrounds the stable focus f2 and is generated by a homoclinic bifurcation [54] to thesaddle s2. As a consequence, the model can achieve the phenomenon of multistability because of the existence of fourattracting sets in the first quadrant: the stable limit cycle Γ1, the origin p0 and the two foci f1 and f2.


Finally, it follows that, in an open subset of parameter space, the (one-dimensional) stable manifold

W s(s1) = {x ∈ A | ϕt(x) −→ s1 as t −→ ∞}

of s1 is the (outer) boundary of the basin of attraction of Γ1. In particular, the lower branch of W s(s1) emerges as the Alleethreshold itself in the phase plane. An orbit beyond this curve converges to Γ1 and implies the survival of the two species.

3.3. Transition between weak and strong Allee effects

A third case is possible in our model; we refer to such a situation as a ‘transition’ from the weak to strong Allee effectand it occurs when L = 0. In this case, if b < k, the origin is repulsive in the x-axis as in a weak Allee effect. However, inthe interior of the first quadrant, p0 has a parabolic attracting sector and a hyperbolic sector. These two regions are dividedlocally by the attracting separatrix

γ =(x, y) ∈ A | y = x + O(x2)

.

Notice that γ represents a critical curve that has to be crossed downwards in the phase plane in order to have a chance ofsurvival. In other words, γ is an Allee threshold, which is typical in the presence of a strong Allee effect; see [23] for moredetails.

4. Preliminary results for the stochastic model

The effect of the random noise in the system (3) becomes stronger in a sufficiently small neighborhood of the curvesP−1(0) ∪ Q−1(0). In fact, in these neighborhoods, by Itô’s rule, the dynamical system behaves like

z(t) ≈ z0 exp

σW (t) −σ 2t2

,

where z stands for x or y, and W (t) is the corresponding Brownian motion with intensity σ .Furthermore, since the stochastic perturbations in (3) are proportional to the population sizes, there exists a sufficiently

small neighborhood of the origin in the first quadrant such that the stochastic trajectories tend to have fluctuations withsmaller amplitudes and to ‘look like’ the deterministic solutions.

On the other hand, it is clear that our stochastic model does not have stationary points, that is, there is no point (x∗, y∗)∈ A, such that F(x∗, y∗) = 0 and G(x∗, y∗) = 0.

In the following Lemmas,wewill show that the solutions of (3) have sample pathwise uniqueness and boundedmoments.This is due to a certain one-sided Lipschitz condition satisfied by the equation coefficients as proven in Lemma 1. In whatfollows, ⟨·, ·⟩ denotes the Euclidean inner product and | · | denotes both the Euclidean norm or the Frobenius matrix norm.

Lemma 1. For the vector field F and the matrix G in Eq. (3) there exist constants λ, ν > 0 such that

⟨a − b, F(a) − F(b)⟩ ≤ λ|a − b|2, ∀a, b ∈ A, (4)

|G(a) − G(b)|2 ≤ ν|a − b|2, ∀a, b ∈ A. (5)

Proof of Lemma 1. Let a = (a1, a2), b = (b1, b2) ∈ A and suppose initially the case a1 > b1. Then, since P(x, y) ≤ rx, forall (x, y) ∈ A, we have:

(a1 − b1)P(a1, a2) ≤ (a1 − b1)ra1,(a1 − b1)P(b1, b2) ≤ (a1 − b1)rb1.

Subtracting both equations leads to:

(a1 − b1)[P(a1, a2) − P(b1, b2)] ≤ r(a1 − b1)2. (6)

It is straightforward to see that inequality (6) is also valid for cases a1 ≤ b1, after subtracting the inequalities:

(b1 − a1)P(b1, b2) ≤ (b1 − a1)rb1,(b1 − a1)P(a1, a2) ≤ (b1 − a1)ra1.

In particular, if a1 = b1, (6) is satisfied in a trivial way.Analogously, since Q (x, y) ≤ sy, for all (x, y) ∈ A, we have:

(a2 − b2)[Q (a1, a2) − Q (b1, b2)] ≤ s(a2 − b2)2. (7)

Adding (6) and (7) we get inequality (4), with λ = max{r, s}. Finally, inequality (5) is immediate, because the coefficientsof G are linear polynomials in x and y, and so, globally Lipschitz functions. �


Lemma 2. For any initial condition X0 ∈ A, system (3) has a unique solution X(t) such that X(0) = X0.Proof of Lemma 2. Consider the translation given by

T : R2−→ R2, such that (x, y) → (x − x0, y − y0), (8)

with (x0, y0) ∈ A. LetF be the newvector fieldC∞-conjugated to F . Notice thatT translates the point (x0, y0) onto the originof the new coordinates and the equivalent vector field satisfiesF(x, y) = F(x + x0, y + y0), for all (x, y) ∈ A := T −1(A). If0 := (0, 0) ∈ A, then from Lemma 1, for all a ∈ A we have:

⟨F(a), a⟩ = ⟨F(a) −F(0), a⟩ + ⟨F(0), a⟩≤ λ|a|2 + |F(0) ∥ a|

≤12|F(0)|2 +

λ +

12

|a|2,

and

|G(a)|2 ≤ 2|G(0)|2 + 2|G(a) −G(0)|2 ≤ 2|G(0)|2 + 2ν|a|2,

whereG(x, y) = G(x + x0, y + y0), with (x, y) ∈ A. Hence:

max⟨F(a), a⟩, |G(a)|2

≤ α + β|a|2, ∀a ∈ A, (9)

where α = max 12 |F(0)|2, 2|G(0)|2

and β = max

λ +

12

, 2ν

. In this way, the existence and uniqueness of solutions

of system (3) follow directly from Theorem 2.3.5 in [55], the C∞-conjugacy (8) and inequality (9). �

It must be stressed that the uniqueness in Lemma 2 is in a pathwise sense. In other words, we say that two solutions X(t)andX(t) of (3) are the same if they have, almost surely, the same sample trajectories for t ∈ [0, T ], i.e.,

P

sup0≤t≤T

|X(t) −X(t)| > 0

= 0.

Lemma 3. For every p > 2 and for any initial point X0 ∈ A there exists C = C(p, T ) > 0 such that the solution X(t) of (3)satisfies:

Esup

0≤t≤T|X(t)|p

≤ C

1 + E[|X0|

p].

Proof of Lemma 3. The statement follows from the one-sided Lipschitz condition in Lemma 1 above and from Lemma 3.2in [29]. �

5. Numerical simulations

In order to study numerically the qualitative behavior of the solutions of (3), given a step ∆t > 0 we use the Split-StepBackward Euler (SSBE) scheme – studied comprehensively in [29] – which is defined as:

SSBEY ∗

k = Yk + F(Y ∗

k ) ∆t;Yk+1 = Y ∗

k + G(Y ∗

k ) ∆Wk.(10)

The scheme (10) calculates approximations Yk ≈ X(tk) for (3), with tk = k∆t , by fixing Y0 = X0, where ∆Wk = W (tk+1)− W (tk) is a normally distributed random variable with zero mean and variance ∆t , for every k = 0, 1, 2, . . . . The methodSSBE emerges as a useful, reliable numerical tool that produces approximations of (3) that converge to the theoreticalsolutions, in the sense presented in the next theorem. This result is a direct consequence of Theorem 3.3 in [29] and theone-sided Lipschitz condition proven in Lemma 1 in this paper.

Theorem 4. Let X(t) be the solution of (3). Then, the numerical solution Yk by the method SSBE (10) applied to (3) has acontinuous-time extension Y(t) (i.e., Y(tk) = Yk) for which

lim∆t→0

Esup

0≤t≤T|Y(t) − X(t)|2

= 0.

Remark. Under the additional hypothesis that the vector field F has polynomial-type growth in its entire domain, Highamet al. prove in [29] that the SSBE method has a convergence rate of order α = 1. However, this type of growth fails to occurin (3) near the origin. Nevertheless, finding the convergence rate and other numerical features of the SSBE method appliedto either system (3) or more general sets of equations is a challenge beyond the scope of this paper.

With the help of the SSBE scheme, we will analyze two biologically significant cases, namely, when the system presentsweak and strong Allee effects, respectively.


a b

dc

Fig. 3. Numerical simulations of system (3) with weak Allee effect with (a) σ1 = σ2 = 0; (b) σ1 = σ2 = 0.01; (c) σ1 = σ2 = 0.1; and (d) σ1 = σ2 = 1.The other parameters are fixed as in Fig. 1. As one ‘turns on’ the noise, the point p ceases to be an equilibrium. Moreover, every trajectory remains in theinterior of the first quadrant, hence suggesting stochastic persistence of the populations.

5.1. The case of a weak Allee effect

In this situation we consider the same parameter values as in Fig. 1. Hence, the original deterministic predation model(1) presents a weak Allee effect, with long term survival of both populations. Moreover, the only equilibrium in the interiorof the first quadrant is the globally attractor node p, see the discussion in Section 3.1. By means of the SSBE method (10) wesimulate trajectories of the extended stochastic predator–prey system (3) for different initial conditions and perturbationintensities σ1, σ2 ∈ [0, 1].

Fig. 3 showsnumerical simulations of these scenarios for increasing values of the environmental noise. In particular, panel(a) shows the same situation as in Fig. 1, i.e., σ1 = σ2 = 0. Notice that for all simulated data there is the long term survivalof populations. This is because the stochastic noise in our model is multiplicative, i.e., proportional to the population sizes.As the prey and predator densities decrease, the amplitude of their fluctuations becomes smaller and the orbits ‘echo’ thedeterministic behaviorwhich never tends towards the origin. Hence, the numerical evidence suggests that there is stochasticpersistence, that is, long term survival with probability one of both species.

Notice also that, in Fig. 3(b)–(d), the point p is no longer an equilibrium of (3). Nevertheless, every trajectory sufficientlynear p always returns in a finite time to a neighborhood of that point. Moreover, it is clear that every orbit in the interior ofthe first quadrant enters a vicinity of p at some instant. This suggests that, for these parameter values, the system is globallyasymptotically stable in mean in the interior of the first quadrant, i.e., in the long term, all the trajectories in A are expectedto approach p [24].

5.2. The case of a strong Allee effect

In this case we consider the same parameter values as in Fig. 2. Hence, the system (3) presents a strong Allee effect.Furthermore, as commented in Section 3.2, there are three closed trajectories corresponding to the periodic solutions Γ1,Γ2 and Γ3, and the origin p0 has an attracting parabolic sector, see Fig. 2. Extending our analysis to the stochastic model (3),we again assume that the intensities of the random noise have bounded values in σ1, σ2 ∈ [0, 1]. In order to illustrateour results, the numerical approximations obtained with the SSBE method (10) are shown in Figs. 4 and 5, where theenvironmental noise affects with equal and different intensities both populations, respectively. In particular, Fig. 4(a) showsthe same situation as in Fig. 2, i.e., σ1 = σ2 = 0.


a

c

e

b

d

f

Fig. 4. Numerical simulations of system (3) with strong Allee effect for (a) σ1 = σ2 = 0; (b) σ1 = σ2 = 0.0001; (c) σ1 = σ2 = 0.001; (d) σ1 = σ2 = 0.01;(e) σ1 = σ2 = 0.1; and (f) σ1 = σ2 = 1. The other parameters are fixed as in Fig. 2. The Allee threshold remains present for different intensities of noise.

With these parameter values, the simulations predict that the closed orbits do not persist, even under relatively smallrandom perturbations. For instance, Fig. 4(b) shows that the global limit cycle Γ3 observed in Fig. 4(a) (for the deterministiccase) has disappeared. Moreover, for small random noises, the corresponding basins of attraction become mere trappingregions that tend to disappear later for larger values of σ1 and σ2; see the sequence of panels (b)–(f) of Fig. 4.

Although the invariant manifoldW s(s1) no longer exists, there is still an Allee threshold present in the system. Indeed, ifthe environmental noise affects both populations in equal measure (i.e., σ1 = σ2) as in Fig. 4, the strong Allee effect remainspresent, showing a threshold in population sizes that has to be exceeded to increase the population sizes. However, thesecritical Allee values are no longer deterministic. Furthermore, the environmental randomness emerges as a vital ingredientto determine the survival or extinction of both species, even with large random noises. For instance, in Fig. 4(f), if initiallythere is a relatively low density of both populations, the model predicts that extinction will not necessarily occur, thoughit may happen in the long term for higher initial densities. In such cases of high noise intensity, the trajectories that do notconverge to the origin instead fluctuate in a neighborhood of point f2, which was an attracting equilibrium in the originaldeterministic system.


a b

Fig. 5. Numerical simulations of system (3) with strong Allee effect with (a) σ1 = 1, σ2 = 0; and (b) σ1 = 0, σ2 = 1. Here the environmental noise affectsthe prey and predator with different intensities; nevertheless, all the trajectories tend to the origin leading both populations to extinction.

On the other hand, if the environmental noise affects the prey and predator with different intensities, that is, σ1 = σ2(as shown in Fig. 5), our model indicates that the extinction as a unique asymptotic equilibrium is possible. In particular,when the randomness affects only one population, the simulations predict that there can be extinction for both species forany initial condition. For instance, all the realizations of the system (3) in Fig. 5 tend to the origin in the long term.

6. Parameter estimation

Usually, the vector of parameters θ :=r, a, b, k,m, n, q, s, σ 2

1 , σ 22

is unknownand itmust be estimated. To this purpose,

we derive a sequence of maximum likelihood estimators (in short MLE) in the discretization of system (2) given by theregular Euler approximation [24].

The maximum likelihood estimation begins with writing a mathematical expression known as the likelihood function ofthe sample data. The likelihood of a set of data is the probability of obtaining that particular set of data, given the chosenprobability distribution model. This expression contains the unknown model parameters θ . The MLE are then the valuesθ =

r, a, b, k, m, n, q, s, σ 2

1 , σ 22

Nof the parameters that maximize the sample likelihood.

In the discrete-time system, our observations are the pair of prey and predator densities Xj, Yj, given byXtj+1 = Xtj +

r1 −

Xtj

k

Xtj −

mXtj

Xtj + b−

qXtjYtj

X2tj + a

∆t + σ1Xtj

W1(tj+1) − W1(tj)

;

Ytj+1 = Ytj +

s

1 −

Ytj

nXtj

Ytj

∆t + σ2Ytj

W2(tj+1) − W2(tj)

.

(11)

Thereforewe look for theMLE θ =

r, a, b, k, m, n, q, s, σ 2

1 , σ 22

Nof the vector of parameters θ for the discrete stochastic

predator–prey model with the Allee effect (11). At time tN , that is, given past observations {(xj, yj), j = 1, 2, . . . ,N}, theMLE is the value of θ that maximizes the likelihood function defined as

L (θ) :=

Nj=1

fxj, yj|xj−k, yj−k : k = 1, 2, . . .

=

Nj=1

fxj, yj|xj−1, yj−1

,

where fx, y |xj−k, yj−k : k = 1, 2, . . .

is the conditional density at point (x, y) of the random variable (xj, yj) given the prior

random variables

{(xj−1, yj−1), . . . , (x1, y1), (x0, y0), (x−1, y−1), . . . , (x−k, y−k), . . .}.

By the specifications (2) and (11), it is clear that zj = (xj, yj) is conditionally distributed as a normal bivariate vector∼N

µj, Σj

given {zj−k : k = 1, 2, . . .}, where Σj is explicitly given as a function of these past observations. Hence by

the Markov property

l(θ) := ln L (θ) = −N2

ln(2π) −N2

ln(σ 21 σ 2

2 ) −12

Ni=1

xi − µxi

σ 21 x

2i

+yi − µyi

σ 22 y

2i

,

where

µxi = xi + r1 −

xik

xi − m

xixi + b

− qxiyi

x2i + a, µyi = yi + s

1 −

yinxi

yi − q

xiyix2i + a

.


Table 1Estimated parameters with initial conditions X(0) = 0.8; Y (0) = 0.8 and time step ∆T = 0.5.

N σ1 = 0.0001 σ2 = 0.0001 q = 4.05116 s = 1

200 6.6448e−005 7.2322e−005 3.8524 1.60381,000 8.1072e−005 8.7575e−005 3.9830 1.0379

10,000 8.0850e−005 9.1729e−005 4.0525 0.9984


N r = 4.02708 a = 3 b = 1 k = 6.79211 m = 4.07697 n = 1

200 4.0068 3.3529 0.1868 6.5893 4.0671 1.93551,000 4.0335 2.9475 0.9797 6.8593 3.9932 1.0020

10,000 4.0507 3.0119 0.9847 6.8072 4.0800 1.0066


N σ1 = 0.01 σ2 = 0.01 q = 1 s = 1

400 3.2267e−004 3.2440e−004 0.8478 1.43881000 7.1885e−004 6.1475e−004 0.8894 0.96045000 1.1879e−003 9.1653e−004 0.99122 0.9888


N r = 2 a = 1 b = 0.5 k = 1 m = 1 n = 1

200 2.6442 1.6287 0.7489 1.8767 −0.4213 2.36171000 1.9791 1.0314 0.5080 0.9972 1.0096 1.19815000 1.9786 0.9985 0.4977 1.0005 0.9980 0.992

We can now easily calculate ∂ l∂σ 2

1and ∂ l

∂σ 22. Then the MLE for the parameters σ 2

1 and σ 22 satisfies:

σ 21 =

1N

Ni=1

(xi − µxi)2

x2i,

σ 22 =

1N

Ni=1

(yi − µyi)2

y2i.

(12)

Analogously, for the rest of the parameters we have the following implicit equations:

∂ l∂k

= 0 ⇒ ˆµxi = x,∂ l∂m

= 0 ⇒

Ni=1

(xi − µxi)

xi(xi + b)= 0,

∂ l∂b

= 0 ⇒

Ni=1

(xi − µxi)

xi(xi + b)2= 0,

∂ l∂q

= 0 ⇒

Ni=1

(xi − µxi)yixi(xi + a)2

= 0,

∂ l∂a

= 0 ⇒

Ni=1

(xi − µxi)yixi(xi + a)2

= 0,∂ l∂r

= 0 ⇒1N

Ni=1

µxi

xi= 1,

∂ l∂n

= 0 ⇒

Ni=1

µyi

xi=

Ni=1

yixi

,∂ l∂s

= 0 ⇒1N

Ni=1

µyi

yi= 1.

(13)

Therefore θ is the solution of the system of eight equations given by (12)–(13), for fixed number of observations N .Wehave implemented this estimator on a standard personal computing platform (PC), andhave observed that it performs

well using simulated data as can be seen from the simulated data in Fig. 3 given in Section 5.1. Despite the apparent algebraiccomplexity of Eqs. (12) and (13) that one needs to solve in order to obtain θ , the problem poses no difficulty for standardsymbolic algebra packages. Using MATLAB’s simulations and algebraic capabilities (Version 7.0) yielded the best computingtimes. In our implementation, which performs an iteration of the algorithm from i = 0 to i = N , values are given with 4significant digits. With initial conditions (X(0), Y (0)) = (0.8, 0.8) and time step ∆T = 0.5, the estimates are shown inTables 1 and 2. On the other hand, Tables 3 and 4 show the estimates with initial points (X(0), Y (0)) = (0.02, 0.01) andtime step ∆T = 0.01. In both cases, the convergence as N increases seems quite fast; N = 1000 is a reasonable number ofdata points for the precision attained above.


7. Discussion

In this work we present the analysis of a model (3) which is a stochastic extension of the predation model with Alleeeffect on prey (1), whose deterministic dynamics has already been studied in [23,10]. We prove that the solutions of thesystem (3) exist, are unique in a pathwise sense and have bounded moments; see Section 4.

Two biologically relevant cases have been studied via numerical simulations, namely, when the prey exhibits a weak andAllee effect, respectively. In the first case (see Section 5.1), we notice that for all the values of the stochastic noise simulated,there is evidence of stochastic persistence of the system, that is, survival of both populations with probability one. Theproperty of non-extinction is then ‘preserved’ from the original deterministic model. In this way, the trajectories fluctuatein the long term around a point in the interior of the first quadrant, this point being the only equilibrium of the originalsystem. Therefore, the numerical evidence suggests that the system (3) has global asymptotical stability in mean.

In the case of the strongAllee effect (see Section 5.2) ourmodel predicts that low randomperturbations canbreak the limitcycles that are present in the deterministic system. On the other hand, there is evidence that the random terms are causinga change in the Allee threshold, which is no longer a fixed deterministic curve. In this respect, for high noise intensities,it has been observed that the orbits may either converge to the origin causing extinction, or fluctuate around the point f2whichwas originally an attracting equilibrium in the deterministic model; the latter case ensures the survival of the species.However, if the environmental randomness affects each population with different intensities, then our model predicts thatthere may be extinction for both species.

In all previous cases, for sufficiently low populations, the trajectories of the stochastic system behave like the ones fromthe original deterministic model, that is, the variability of the solution process dies down as the populations decrease. Thisis because the stochastic noises have been considered as proportional to the population sizes. This consideration impliesthat, for this type of model, the larger consequences of the inclusion of environmental randomness become evident just forsufficiently high population sizes. The numerical investigations of the model were performed with the SSBE method (10),which ensures that our approximations converge to the theoretical solutions; see Theorem 4. Nevertheless, from a rigorousmathematical perspective, results on the corresponding convergence rates of this numerical scheme applied to (3) remaina considerable challenge beyond the scope of this paper.

Finally, for the Euler discretization of the model, the maximum likelihood estimator of the parameters can be found ina practical way. This allows one to estimate the value of all the parameters in the model provided one has a sufficientlylarge number of data for prey and predator densities. In fact, numerical results based on simulations show that the methodperforms well (see Section 6).

Since the Allee effect and environmental noise can interact due to a wide range of biological phenomena, the recognitionof their consequences on the reproduction, conservation and behavior of species emerges as an important goal for thepopulation dynamics community. Our approach of considering multiplicative-type random noise in a systemwith the Alleeeffect can be seen as a first step in the study of more general models. In this sense, potential future research involves theanalysis of more general stochastic perturbations, which may even include considering random parameters (e.g., stochasticratio-dependent models), and different ways of modelling the Allee effect.

Acknowledgments

The preparation of this manuscript was partially done while PA was affiliated to the University of Bristol in the UnitedKingdom. PAwas also partially funded by Proyecto Basal CMMUniversidad de Chile. EGOwas partially funded by FONDECYT1120218 and DIEA-PUCV 124.730/2012. ST was partially supported by Proyecto Anillo ACT/12 Red de Análisis Estocástico yAplicaciones, Prosul - Brazil, and DIPUV-REG 26/2009.

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stochastic predator–prey model with allee effect on prey

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