Persistence in a stochastic intraguild predation model

Applied Mathematics Letters 63 (2017) 59–64

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Applied Mathematics Letters www.elsevier.com/locate/aml

Persistence in a stochastic intraguild predation model✩ Jiangtao Yang, Wendi Wang ∗ Key Laboratory of Eco-environments in Three Gorges Reservoir Region, School of Mathematics and Statistics, Southwest University„ Chongqing 400715, China

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Article history: Received 7 May 2016 Received in revised form 22 July 2016 Accepted 22 July 2016 Available online 29 July 2016

abstract A mixed competition–predation model is proposed where the switches between competition and predation are seasonal-dependent. The threshold values are derived above which the populations are stochastically permanent, and below which a population becomes extinct for the prey–predator interactions or the weak competitions. © 2016 Elsevier Ltd. All rights reserved.

Keywords: Stochastic perturbation Stochastic persistence Extinction Mixed interaction

1. Introduction A number of papers [1,2] have recently considered the stochastic predator–prey model with time periodic coefficients:  dx1 (t) = x1 (t)[(r1 (t) − a11 (t)x1 (t) − a12 (t)x2 (t))dt + σ1 (t)dB1 (t)], (1.1) dx2 (t) = x2 (t)[(r2 (t) + a21 (t)x1 (t) − a22 (t)x2 (t))dt + σ2 (t)dB2 (t)], where xi (t) is the size of the ith population at time t and Bi (t) is the Brownian motion; the growth rates, intra-specific regulation coefficients, predation coefficient and conversion coefficient of populations are denoted by a11 , a22 , a12 and a21 , respectively; σi represents the noise amplitude. Liu et al. [1] obtained sufficient conditions for extinction and persistence in the mean of system (1.1). Zu et al. [2] established sufficient conditions for the existence of a positive periodic solution in system (1.1). In many ecological systems, one or more species may act as both predator and competitor with other species at the same or similar trophic level. This phenomenon is termed as intraguild predation [3]. For example, largemouth bass (Micropterus salmoides) is native to North America. Its juveniles compete with ✩

The research is supported by the National Natural Science Foundation of China (No. 11571284). ∗ Corresponding author. E-mail address: [email protected] (W. Wang).

http://dx.doi.org/10.1016/j.aml.2016.07.022 0893-9659/© 2016 Elsevier Ltd. All rights reserved.

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small fish, but adults are primarily piscivorous, and consume a wide variety of small-bodied and juvenile fish when food is scarce in summer [4]. Therefore, largemouth basses are competitors and predators of small fish depending on time. Motivated by the above biological contexts, in this paper we assume that ri (t), σi (t) and aij (t) (j = 1, 2) are continuous and periodic functions with period T > 0. Further, a11 (t), a12 (t) and a22 (t) are positive functions, whereas a21 can change sign to describe the switches of population x2 between competitor and predator. The main aim of this paper is to investigate the stochastic persistence and extinction of system (1.1). The threshold conditions of stochastic persistence of system (1.1) are established, which improves and extends the corresponding results in [1,2]. 2. Stochastic persistence and extinction Throughout this paper, we denote by (Ω , F, {Ft }t≥0 , P ) a complete probability space with a filtration {Ft }t≥0 satisfying the usual conditions (see [5]). Let B1 and B2 denote the independent standard Brownian motions defined on this probability space. In view of the biological significance of the model, we discuss the 2 2 dynamics of system (1.1) in R+ = {(x1 , x2 ) ∈ R2 : x1 ≥ 0, x2 ≥ 0} and its interior IntR+ . For convenience t T u l T we let f = maxt∈[0,T ] f (t), f = mint∈[0,T ] f (t), ⟨f (t)⟩ = 0 f (s)ds/t and f = 0 f (s)ds/T , where f (t) is a continuous T -periodic function. Let z(t) = (x1 (t), x2 (t)) denote a solution of system (1.1) corresponding 2 . By similar discussions to those in [6,7], we can obtain the to an initial function z0 = (x1 (0), x2 (0)) ∈ R+ following lemma. 2 Lemma 2.1. For any ε ∈ (0, 1), there is a positive constant H = H(ε) such that for any initial data z0 ∈ R+ , the solution z(t) = (x1 (t), x2 (t)) of (1.1) exists for all t ≥ 0 and satisfies

lim sup P {x1 (t) > H(ε)} < ε, t→∞

lim sup P {x2 (t) > H(ε)} < ε.

(2.1)

t→∞

We need to consider the subsystem of (1.1): dφi (t) = φi (t)[(ri (t) − aii (t)φi (t))dt + σi (t)dBi (t)],

φi (0) = xi (0),

i = 1, 2.

(2.2)

Let bi (t) := ri (t) − σi2 (t)/2. By Theorem 2.1 in [8], we can state the following lemma. Lemma 2.2. Assume bT1 > 0 and bT2 > 0. Let (φ1 (t), φ2 (t)) be the solution of system (2.2) with initial 2 value (x1 (0), x2 (0)) ∈ IntR+ . Then there exists a unique positive T -periodic solution (φ∗1 (t), φ∗2 (t)) of system (2.2) such that limt→∞ (|φ1 (t) − φ∗1 (t)| + |φ2 (t) − φ∗2 (t)|) = 0 a.s. Set λ1 (t) = b1 (t) − a12 (t)φ∗2 (t),

λ2 (t) = b2 (t) + a21 (t)φ∗1 (t),

where φ∗1 (t) and φ∗2 (t) are defined in Lemma 2.2. 2.1. Stochastic persistence

Theorem 2.1. Assume that bT2 > 0 holds. If λT1 > 0 and λT2 > 0 a.s., then system (1.1) is stochastically permanent.

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Proof. It is easy to see that ⟨λi (t)⟩ → λi T as t → ∞ with probability one (see [9]). Let us choose positive constants k0 and ε such that for any continuous function ν(t) satisfying |ν(t) − φ∗1 (t)| < ε/2 for all t ≥ k0 T , we have λT1 − (au12 + au11 )ε > 0,

lim sup⟨b2 (t) + a21 (t)ν(t)⟩ − au22 ε > 0

a.s.

(2.3)

lim inf P {x2 (t) ≤ H(ε)} ≥ 1 − ε.

(2.4)

t→∞

From (2.1), for any ε ∈ (0, 1) we have lim inf P {x1 (t) ≤ H(ε)} ≥ 1 − ε, t→∞

t→∞

Set Ωtε := {ω ∈ Ω : x1 (t, ω) ≤ H(ε), x2 (t, ω) ≤ H(ε)},

ε Ω∞ := {ω ∈ Ω : lim inf P {Ωtε } ≥ 1 − ε}. t→∞

ε ) P (Ω∞

≥ 1 − ε. Then for any ϵ0 ∈ (0, 1 − ε), there is a positive integer k1 > k0 such that By (2.4), we have P (Ωtε ) ≥ 1 − ε − ϵ0 for all t ≥ k1 T . Consequently, for every ω ∈ Ωtε , we have 0 ≤ x1 (t, ω) ≤ H(ε),

0 ≤ x2 (t, ω) ≤ H(ε),

for all t ≥ k1 T.

(2.5)

2 Now we choose α(ε), β(ε) in (0, ε) and let Dε ⊂ IntR+ be the region enclosed by x1 = α(ε), x2 = β(ε), x1 = H(ε) and x2 = H(ε). We show below that lim inf t→∞ P {z(t) ∈ Dε } ≥ 1 − ε.

Step 1: We first show that there is a constant β1 (ε) > 0 such that lim sup x2 (t, ω) > β1 (ε) t→∞

ε for any ω ∈ Ω∞ .

(2.6)

For any constant β1 > 0, we consider the following equation: dX1 (t) = X1 (t)[(r1 (t) − a11 (t)X1 (t) − 2a12 (t)β1 )dt + σ1 (t)dB1 (t)].

(2.7)

T Since 0 (b1 (s) − 2a12 (s)β1 )ds/T > 0 for sufficiently small β1 , it follows from Lemma 2.2 that Eq. (2.7) has a unique positive T -periodic solution Xβ∗1 (t) which is globally asymptotically stable. It is obvious that Xβ∗1 (t) → φ∗1 (t) uniformly in [0, T ] as β1 → 0. Hence, there is constant β1 = β1 (ε) > 0 and 2β1 < ε such that Xβ∗1 (t) > φ∗1 (t) − ε/2

for all t ∈ [0, ∞) a.s.

(2.8)

If (2.6) is not true, there exist a solution (x1 (t), x2 (t)) with positive initial values x1 (0) > 0 and x2 (0) > 0, a non-empty set Ω1 ⊂ Ωtε with P (Ω1 ) > 0, and a positive integer k2 > k1 such that x2 (t, ω) < 2β1 for all ω ∈ Ω1 and t ≥ k2 T . Note that dx1 (t) ≥ x1 (t)[(r1 (t) − a11 (t)x1 (t) − 2a12 (t)β1 )dt + σ1 (t)dB1 (t)] for all ω ∈ Ω1 and t ≥ k2 T , and dx1 (t) ≤ x1 (t)[(r1 (t) − a11 (t)x1 (t))dt + σ1 (t)dB1 (t)]. Then, by (2.8), Lemma 2.2 and the comparison theorem of stochastic equation [10], there is a positive integer k3 > k2 such that |x1 (t) − φ∗1 (t)| < ε for all t ≥ k3 T and ω ∈ Ω1 . According to the second equation of (1.1), we have   ln x2 (t) ln x2 (k3 T ) 1 t 1 t ≥ + (b2 (s) + a21 (s)x1 (s) − 2au22 β1 ]ds + σ2 (s)dB2 (s) t t t k3 T t k3 T

(2.9)

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for all t ≥ k3 T . Therefore, by the strong law of large numbers for martingales (see, e.g. [5]), we get lim sup t−1 ln x2 (t) ≥ lim sup⟨b2 (t) + a21 (t)x1 (t)⟩ − 2au22 β1 , t→∞

t→∞

which implies that x2 (t) → ∞ as t → ∞, which contradicts (2.5). Hence, Step one is proved. Step 2: In this step we show that there is a constant β(ε) > 0 such that ε lim inf P {ω ∈ Ω∞ : β(ε) < x2 (t, ω) ≤ H(ε)} ≥ 1 − ε. t→∞

(2.10)

Indeed, if we can prove that there is a constant β(ε) > 0 such that lim inf x2 (t, ω) > β(ε) t→∞

ε for any ω ∈ Ω∞ ,

then (2.10) follows immediately. Suppose that the conclusion is not true. Then there exist a sequence 2 and a non-empty set Ω2 ⊂ Ωtε , P (Ω2 ) > 0, and a positive integer k3 (k3 > k1 ) such that {zn } ⊂ IntR+ inf t≥k3 T x2 (t, zn , ω) < β1 (ε)/(n + 1) for all ω ∈ Ω2 , where (x1 (t, zn ), x2 (t, zn )) is a solution with initial value zn . According to (2.5), we have dE[ln x2 (t, zn )IΩ2 ] ≥ [b2 (t) − (|a21 (t)| + a22 (t))H(ε)]P (Ω2 ) (2.11) dt for all t ≥ k3 T . By Step one, there exist a non-empty set Ω3 ⊆ Ω2 , P (Ω3 ) > 0, and positive integers k4 , k5 (k5 > k4 > k3 ) such that x2 (k4 T, zn ) ≥ β1 (ε)

and x2 (k5 T, zn ) ≤ β1 (ε)/(n + 1)

for all ω ∈ Ω3 . Set τn = inf{t : x2 (t, zn ) > β1 (ε)/(n + 1), ω ∈ Ω3 , t > k4 T }, τ = sup{t < τn : x2 (t, zn ) < β1 (ε), ω ∈ Ω3 , t > k4 T }. Then, x2 (τn , zn ) = β1 (ε)/(n + 1), x2 (τ, zn ) = β1 (ε) and β1 (ε)/(n + 1) < x2 (t, zn ) < β1 (ε) for all t ∈ (τ, τn ) and ω ∈ Ω3 . By integrating (2.11) from τ to τn , we obtain  τn [(|a21 (s)| + a22 (s))H(ε) − b2 (s)]ds ≥ −E[ln x2 (τn , zn )IΩ3 ] + E[ln x2 (τ, zn )IΩ3 ] τ

= ln(n + 1)P (Ω3 ). Thus, we obtain τn − τ → ∞ as n → ∞. By (2.3), we see that there are constants T0 and γ such that for any t ≥ T0 and a ≥ 0,  a+t (b2 (s) + a21 (s)ν(s) − au22 ε)ds > γ a.s.

(2.12)

(2.13)

a

For any t ∈ [τ, τn ] and ω ∈ Ω3 , we have dx1 (t, zn ) ≥ x1 (t, zn )[(r1 (t) − a11 (t)x1 (t, zn ) − a12 (t)β1 (ε))dt + σ1 (t)dB1 (t)] and dx1 (t, zn ) ≤ x1 (t, zn )[(r1 (t) − a11 (t)x1 (t, zn ))dt + σ1 (t)dB1 (t)]. It follows from the proof of Step one that there is a positive integer k6 > T0 /T such that |x1 (t, zn )−φ∗1 (t)| < ε for all t ≥ k6 T + τ and ω ∈ Ω3 . By (2.12), there is a N0 > 0 such that τn > 2k6 T + τ for all n ≥ N0 . Hence, |x1 (t, zn ) − φ∗1 (t)| < ε for all t ∈ [k6 T + τ, τn ] and ω ∈ Ω3 .

(2.14)

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According to the second equation of (1.1), we have 

τn

E[ln x2 (τn , zn )IΩ3 ] ≥ E[ln x2 (k6 T + τ, zn )IΩ3 ] + E



(b2 (s) + a21 (s)x1 (s, zn ) −

au22 β1 (ε))ds

 IΩ3 ,

k6 T +τ

which implies     τn β1 (ε) β1 (ε) 1 u (b2 (s) + a21 (s)x1 (s, zn ) − a22 β1 (ε))ds IΩ3 . ≥ exp E 1+n 1+n P (Ω3 ) k6 T +τ Consequently, by (2.13) we have β1 (ε)/(1 + n) ≥ β1 (ε)eγ /(1 + n) > β1 (ε)/(1 + n), which is a contradiction, and Step two is proved. Using the same arguments as those in Steps one and two, we can obtain lim inf P {α(ε) < x1 (t) ≤ H(ε)} ≥ 1 − ε. t→∞

ε ) ≥ 1 − ε. This completes Summarizing the above analysis, we obtain lim inf t→∞ P {z(t) ∈ Dε } = P (Ω∞ the proof of Theorem 2.1. 

Theorem 2.2. Assume that bT1 > 0 and bT2 < 0 hold. If λT2 > 0 a.s., then system (1.1) is stochastically permanent. The proof of Theorem 2.2 is omitted because it is similar to it for Theorem 2.1. 2.2. Extinction By Lemma 2.2 and the similar arguments to those in [11], we obtain the following theorems. Theorem 2.3. (1) Assume that bT2 > 0 and al21 ≥ 0 hold. If λT1 < 0 a.s., then population x1 is extinct with probability one and population x2 is stochastically permanent; (2) Assume that bT1 > 0 and al21 ≥ 0 hold. If λT2 < 0 a.s., then population x2 is extinct with probability one and population x1 is stochastically permanent. Theorem 2.4. (a) If bT1 < 0 and bT2 > 0, then population x1 is extinct with probability one and population x2 (t) is stochastically permanent; (b) If bT1 < 0 and bT2 < 0, then populations x1 and x2 go to extinction with probability one. 3. Examples In this section, we give an example to illustrate our main results in this paper. Fix the parameters by r1 (t) = r1 := 0.8, r2 (t) = −0.5 − 0.1 sin2 (πt), a11 (t) ≡ a11 > 0, a12 (t) = 0.45 + 0.4 sin2 (πt), a22 (t) = 0.3 + 0.1 sin2 (πt), σ1 (t) = 0, σ2 (t) = 0.12 + 0.1 sin2 (πt). Then we have bT1 = b1 = 0.8 > 0, bT2 = −0.565075 < 0. We let a21 (t) = −a + sin2 (πt), where a is a nonnegative and adjustable parameter so that a21 can change sign to describe the switches of population x2 between competitor and predator. Case 1. a = 0. Then population x2 acts as predator for population x1 . Note that  1 0.8 0.4 ds = −0.565075 + . λT2 = −0.565075 + sin2 (πt) a a 11 11 0

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By Theorems 2.2 and 2.3-(2), we have (C1) If a11 > 0.7079, i.e., λT2 < 0, then population x2 is extinct with probability one and population x1 is stochastically permanent; (C2) If a11 < 0.7079, i.e., λT2 > 0, then system (1.1) is stochastically permanent. In comparison, Liu et al. in [1] showed that the conclusions in (C1) hold if a11 > 1.4157. Furthermore, Zu et al. in [2] concluded that the system is persistent in mean, and has at least one positive T -periodic solution if bT2 + al21 b1 /a11 > 0, which is not satisfied because al21 = 0 and bT2 = −0.565075 < 0 for this example. Case 2. a ∈ (0, 1). Then population x2 behaves as predator for t ∈ [nπ + d1 , nπ + d2 ], n = 0, 1, 2, . . . , and competes with population x1 for other times, where d1 , d2 ∈ (0, 1) with d1 < d2 such that a21 (d1 ) = a21 (d2 ) = 0. Case 3. a ≥ 1. Then population x2 competes with population x1 for all times. In the last two cases, by Theorem 2.2, λT2 > 0 implies that system (1.1) is stochastically permanent, which extends the corresponding results in [1,2] to the mixed competition–predation case or the pure competition case. References [1] M. Liu, K. Wang, Persistence, extinction and global asymptotical stability of a non-autonomous predator–prey model with random perturbation, Appl. Math. Model. 36 (2012) 5344–5353. [2] L. Zu, D. Jiang, D. O’Regan, B. Ge, Periodic solution for a non-autonomous Lotka–Volterra predator–prey model with random perturbation, J. Math. Anal. Appl. 430 (2015) 428–437. [3] J.M. Fedriani, T.K. Fuller, R.M. Sauvajot, E.C. York, Competition and intraguild predation among three sympatric carnivores, Oecologia 125 (2000) 258–270. [4] T.G. Brown, B. Runciman, S. Pollard, A.D.A. Grant, Biological synopsis of largemouth bass (Micropterus salmoides), Can. Mansucr. Rep. Fish. Aquat. Sci. 2884 (2009) 1–27. [5] X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997. [6] P.S. Mandal, S. Abbas, M. Banerjee, A comparative study of deterministic and stochastic dynamics for a non-autonomous allelopathic phytoplankton model, Appl. Math. Comput. 238 (2014) 300–318. [7] K. Wang, Y. Zhu, Dynamics of a stochastic predator–prey model with mutual interference, Int. J. Biomath. 7 (2014) 1450026. [8] B. Zhang, K. Gopalsamy, On the periodic solution of n-dimensional stochastic population models, Stoch. Anal. Appl. 18 (2000) 323–331. [9] R. Khasminskii, Stochastic Stability of Differential Equations, second ed., Springer-Verlag, Berlin, Heidelberg, 2012. [10] N. Ikeda, S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math. 14 (1977) 619–633. [11] M. Liu, K. Wang, Extinction and permanence in a stochastic nonautonomous population system, Appl. Math. Lett. 23 (2010) 1464–1467.