Positive periodic solutions for Lotka–Volterra systems with a general attack rate

POSITIVE PERIODIC SOLUTIONS FOR LOTKA-VOLTERRA SYSTEMS WITH A GENERAL ATTACK RATE CRISTINA LOIS-PRADOS AND RADU PRECUP Abstract. The paper deals with a non-autonomous Lotka-Volterra type system, which in particular may include logistic growth of the prey population and hunting cooperation between predators. We focus on the existence of positive periodic solutions by using an operator approach based on the Krasnosel’skii homotopy expansion theorem. We give sufficient conditions in order that the localized periodic solution does not reduce to a steady state. Particularly, two typical expression for the functional response of predators are discussed. Key words: non-autonomous Lotka-Volterra system, hunting cooperation, logistic growth, periodic solution, existence, localization Mathematics Subject Classification: 34C25, 47J05, 92D25 1. Introduction In this paper we consider non-autonomous Lotka-Volterra type systems with a general attack rate (1.1) (x0=a(t)xg(x)−ϕ(t, x, y)xy y0(t) = −b(t)y+c(t)ϕ(t, x, y)xy, where a, b, c ∈ C(R,R+) are ω-periodic for the same period ω > 0, a, b 6≡ 0, mins∈[0,ω]c(s)>0; ϕ∈ C(R×R+×R+,R+) is such that ϕ(·, x, y) is ω-periodic for every (x, y)∈R+×R+; and g∈ C(R+,R) is decreasing, with g(0) ≤1. In particular, we consider g(x)≡1 (linear growth of the prey), or g(x)≡1−x K(logistic growth of the prey) and one of the following expressions for the attack rate ϕ, ϕI(t, x, y)≡λ(t) + α(t)y, ϕII (t, x, y)≡λ(t) + α(t)y 1 + β(t)(λ(t) + α(t)y)x, both used in the literature to simulate cooperation between predators [12]. Here, we assume that α, β, λ ∈ C(R,R+) are ω-periodic functions and λ, β 6≡ 0. Lotka-Volterra type systems are commonly use to describe interactions between two species, prey and predator. In the autonomous case, these models have a Kolmogorov structure, being of the form (x0=x F(x, y) y0=y G(x, y), 1 2 CRISTINA LOIS-PRADOS AND RADU PRECUP and most of them satisfy the following conditions: Fy(x, y)>0, Gx(x, y)>0 and Gy(x, y)≤0 (see, e.g., [3, Section 5.4]). For non-autonomous Kolmogorov type systems, we refer the reader to the paper by Zanolin [15]. The original model given by Lotka [8] and Volterra [14] is the following one (1.2) (x0=ax −λxy y0=−by +cλxy. For a historical note on this classical model, see [1]. As suggested by Volterra himself, a more realistic prey growth is the logistic one, as is the case of Rosenzweig-MacArthur model [11], namely (x0=ax 1−x K−φ(x)y y0=−by +cφ(x)y. Some generalizations of the Rosenzweig-MacArthur model are given in [6], where in particular, it is consider the logistic growth for both prey and predator populations (see also [4]). As regards the function ϕinvolved in the functional response of predators to the change of densities in (1.1), we can mention the paper by Berec [2], where ϕhas the form (1.3) ϕ(x, y) = λ+αy 1 + β(y)(λ+αy)x, which expresses the effects of hunting cooperation between predators. Also, in a recent paper by Alves & Hilker [12], there are used the particular expressions of ϕI, ϕII involving constant coefficients, namely ϕ(x, y) = λ+αy (λ > 0, α ≥0) and ϕ(x, y) = λ+αy 1 + β(λ+αy)x(β > 0). In this paper, we consider the more general system (1.1), which is non-autonomous, involves a general prey growth g, and a general functional response of predators. We proceed as follows: In Section 2, we give the integral version of the system, we state the Krasnosel’skii type homotopy fixed point theorem, which is our main tool, and we give some useful notations. In Section 3, we first study the steady states of the system giving a necessary and sufficient condition for the existence of positive equilibriums. Then, in Subsection 3.2, we state and prove the main result about the existence and localization of periodic solutions, whose sufficient conditions are particularized for the cases of linear and logistic prey growth. In both situations, the possibility of localization is discussed separately for ϕ=ϕIand ϕ=ϕII . The method that we use is based on a completely different topological argument as compared to the one in [13], where only a particular case of our system is studied. Our proof appears to be simpler and more natural. In Subsection 3.3, we give sufficient conditions in order that the localized periodic solution does not reduce to a steady state, while in Subsection 3.4, a positivity result about nonconstant periodic solutions is included. Next, in Subsection 3.5, the existence results are improved for the case that ϕdoes not depend on time. Finally, the autonomous system is discussed as a very particular case. LOTKA–VOLTERRA TYPE SYSTEMS 3 2. Preliminaries We are interested in ω-periodic solutions of system (1.1) and for this purpose we use an operator approach as in [10] (see also [13]). This approach is based on the fact that for every ω-periodic functions f1, f2∈ C(R,R) and a, b ∈ C(R,R+), a, b 6≡ 0, there is a unique ω-periodic solution (x, y) of the system (2.1) (x0=a(t)x−f1(t) y0=−b(t)y+f2(t), given by (x(t) = Rt+ω tH1(t, s)f1(s)ds y(t) = Rt+ω tH2(t, s)f2(s)ds, where H1(t, s) = e−Rs ta(τ)dτ 1−e−Rω 0a(τ)dτ , H2(t, s) = eRs tb(τ)dτ eRω 0b(τ)dτ −1((t, s)∈R×R). Now, if instead of linear system (2.1), we consider the nonlinear system (x0=a(t)x−f1(t, x, y) y0=−b(t)y+f2(t, x, y), then its ω-periodic solutions are exactly the ω-periodic solutions of the nonlinear integral system (x(t) = Rt+ω tH1(t, s)f1(s, x(s), y(s)) ds y(t) = Rt+ω tH2(t, s)f2(s, x(s), y(s)) ds, that can be studied as a fixed point equation. In this paper, the fixed point arguments are based on the following Krasnosel’skii type result in cones (see, for example, [9, Theorem 10.8]). We recall that by a cone Cin a Banach space X, we mean a closed convex set such that λC ⊂C, for every λ∈R+, and C∩(−C) = {0}. Theorem 2.1. Let (X, k·k)be a Banach space, C⊂Xa cone, 0< r < R and N:CR−→ C a compact operator, where CR={u∈C:kuk ≤ R}. Assume that (E1)N(u)6=λu, for every u∈C,kuk=rand all λ > 1, (E2)there exists v∈C\{0}such that u−N(u)6=λv, for every u∈C,kuk=Rand all λ > 0. Then, Nhas a fixed point uin Cwith r≤ kuk ≤ R. We conclude this preliminary section by the list of notations that are useful to simplify the computations related to the application of Theorem 2.1 to our system (1.1). a:= min s∈[0,ω]a(s), b := min s∈[0,ω]b(s), c := min s∈[0,ω]c(s), a:= max s∈[0,ω]a(s), b := max s∈[0,ω]b(s), c := max s∈[0,ω]c(s), 4 CRISTINA LOIS-PRADOS AND RADU PRECUP m1:= min (t,s)∈[0,ω]×[t,t+ω]H1(t, s) = 1 eRω 0a(τ)dτ −1, m1:= min t∈[0,ω]Zt+ω t H1(t, s)ds, m2:= min (t,s)∈[0,ω]×[t,t+ω]H2(t, s) = 1 eRω 0b(τ)dτ −1, m2:= min t∈[0,ω]Zt+ω t H2(t, s)ds, M1:= max (t,s)∈[0,ω]×[t,t+ω]H1(t, s) = 1 1−e−Rω 0a(τ)dτ , M1:= max t∈[0,ω]Zt+ω t H1(t, s)ds, M2:= max (t,s)∈[0,ω]×[t,t+ω]H2(t, s) = 1 1−e−Rω 0b(τ)dτ , M2:= max t∈[0,ω]Zt+ω t H2(t, s)ds, q1:= m1 M1 , q2:= m2 M2 , m3:= q1q2min{m1, c m2}, M3:= max{M1, c M2}, m3:= q1q2min{m1, c m2}, M3:= max{M1, c M2}. We shall also use the notation kxk∞for the max norm of x∈ C([0, ω],R),i.e., kxk∞= max t∈[0,ω]|x(t)|. 3. Main results 3.1. Steady states. We begin by looking for the steady states of system (1.1), that is for points (x0, y0)∈R+×R+such that x0(a(t)g(x0)−ϕ(t, x0, y0)y0) = 0, y0(c(t)ϕ(t, x0, y0)x0−b(t)) = 0, (3.1) for all t∈R. It is clear that (0,0) is a solution of (3.1). In the following, we distinguish three cases: Case I: x0= 0, y0>0. Under this conditions, the first equation in (3.1) is obviously satisfied, while from the second one we have y0b(t) = 0 for every t∈[0, ω], which is not possible for y0>0 and b6≡ 0. Therefore, there are no steady states of the type (0, y0),with y0>0. Case II: x0>0, y0= 0. Now the second equation in (3.1) trivially holds, while the first one gives a(t)g(x0) = 0 for all t∈[0, ω]. As a6≡ 0, one must have g(x0) = 0. Therefore, a point of the form (x0,0), x0>0 is a steady state if and only if g(x0) = 0. Case III: x0>0, y0>0. Under this situation, system (3.1) is equivalent to ϕ(t, x0, y0) = a(t)g(x0) y0 =b(t) c(t)x0 for every t∈[0, ω]. LOTKA–VOLTERRA TYPE SYSTEMS 5 The conclusions about the steady states of system (1.1) are collected in the following proposition. Proposition 3.1. A point (x0, y0)∈R+×R+is a steady state of system (1.1), if and only if one of the following conditions holds: (a) x0= 0 and y0= 0; (b) x0>0, g(x0)=0and y0= 0; (c) x0>0, y0>0and (3.2) ϕ(t, x0, y0) = a(t)g(x0) y0 =b(t) c(t)x0 for every t∈[0, ω]. According to this proposition, in case of considering the linear growth of the prey population, case (b) is not possible, and steady states of the form (x0, y0) with x0, y0>0 exist if and only if (3.3) ϕ(t, x0, y0) = a(t) y0 =b(t) c(t)x0 for every t∈[0, ω]. If one considers the logistic growth of the prey population, then from case (b) we have the steady state (K, 0), and steady states of the form (x0, y0) with x0, y0>0 exist if and only if (3.4) ϕ(t, x0, y0) = a(t)1−x0 K y0 =b(t) c(t)x0 for every t∈[0, ω]. Coming back to the general system (1.1), let us note that if there is not any constant k > 0 such that (3.5) a(t) = kb(t) c(t)for all t∈[0, ω], then the system has no steady states (x0, y0) with x0, y0>0. Therefore, under condition (3.5), the orbits of all ω-periodic solutions (x, y), with x(t), y(t)>0 for every t∈[0, ω], do not reduce to points. 3.2. Existence of periodic solutions. In this section, we prove the existence of ω-periodic solutions of system (1.1). To this aim, we introduce the following conditions on ϕ: (i) there exists η∈ C(R×R+×R+,R+) such that •η(·, x, y) is ω-periodic for every (x, y)∈R+×R+, •η(t, ·, y), η(t, x, ·) are increasing functions for every (t, y),(t, x)∈R×R+, •ϕ(t, x, y)≤η(t, x, y) for every (t, x, y)∈R×R+×R+, (ii) there exists ψ∈ C(R×R+,R+) such that •ψ(·, z) is ω-periodic for every z∈R+, •ψ(t, ·) is decreasing for every t∈R, •ϕ(t, x, y)≥ψ(t, x +y) for every (t, x, y)∈R×R+×R+. The main result of this paper is the following Theorem 3.1. Let conditions (i) and (ii) hold. If there exist r, R ∈R,0< r < R such that (3.6) 1 ≥a M1(1 −g(r)) , (3.7) M1a(1 −g(r)) + M3rZω 0 η(s, r, r)ds ≤2, 6 CRISTINA LOIS-PRADOS AND RADU PRECUP (3.8) RZω 0 ψ(s, R)ds ≥2 m3 , then system (1.1) has an ω-periodic solution (x, y)∈Csuch that r≤ k(x, y)k=kxk∞+kyk∞≤R. Proof. We apply Theorem 2.1 in the Banach space Xω:= (x, y)∈ C(R,R)2:x(t) = x(t+ω), y(t) = y(t+ω) for every t∈R, endowed with the norm k(x, y)k:= kxk∞+kyk∞ and with the cone C:= {(x, y)∈Xω:x(t)≥q1kxk∞, y(t)≥q2kyk∞for every t∈R}, to the operator N= (N1, N2),where N1(x, y)(t) := Zt+ω t H1(t, s) [a(s)x(s)(1 −g(x(s))) + ϕ(s, x(s), y(s)) x(s)y(s)] ds, N2(x, y)(t) := Zt+ω t H2(t, s)c(s)ϕ(s, x(s), y(s)) x(s)y(s)ds. (3.9) As shown in Preliminaries, the ω-periodic solutions of system (1.1) are the fixed points in Xω of the operator N. First note that since a, c, ϕ, 1−gand H1,H2are nonnegative functions, one has N(C)⊂C. In addition, the compactness of Nimmediately follows from the Arzel`a–Ascoli theorem. It remains to prove that conditions (E1) and (E2) hold, where the element v∈C\{0}is chosen to be any (x0, y0) with x0, y0>0. We start by proving condition (E1), which in our case reads as follows (3.10) (N1(x, y), N2(x, y)) 6=λ(x, y) for every (x, y)∈C, k(x, y)k=rand all λ > 1. To this aim, we consider three cases: (a) Assume x≡0, y6≡ 0.Then condition (3.10) trivially holds since N1(x, y), N2(x, y)≡0. (b) If x6≡ 0 and y≡0, then N2(x, y)≡0 and (3.10) reduces to (3.11) N1(x, 0) 6=λx for all (x, 0) ∈C, kxk∞=rand all λ > 1. To proof this, assume the contrary, namely that there exist (x, 0) ∈C,kxk∞=rand λ > 1 such that N1(x, 0)(t) = λx(t) for every t∈[0, ω]. Let t0∈[0, ω] be such that x(t0) = kxk∞=r > 0, then also using the property that −gis increasing, one has r=x(t0)< λx(t0) = Zt0+ω t0 H1(t0, s)a(s)x(s) (1 −g(x(s))) ds ≤a r (1 −g(r)) Zt0+ω t0 H1(t0, s)ds ≤a r (1 −g(r)) M1. Dividing by r > 0 yields 1 < a (1 −g(r)) M1,which contradicts our hypothesis (3.6). Thus, condition (3.11) holds. LOTKA–VOLTERRA TYPE SYSTEMS 7 (c) Finally, we prove that condition (3.10) holds for x, y 6≡ 0. If it does not hold, then there exists such a pair (x, y)∈C,k(x, y)k=rand λ > 1 with N1(x, y)(t) = λx(t), N2(x, y)(t) = λy(t) for every t∈[0, ω]. Let t0∈[0, ω] be such that kxk∞=x(t0).Then also using condition (i) over ϕ, one has kxk∞=x(t0)< λx(t0) = N1(x, y)(t0) =Zt0+ω t0 H1(t0, s)[a(s)x(s)(1 −g(x(s))) + ϕ(s, x(s), y(s)) x(s)y(s)]ds ≤akxk∞(1 −g(kxk∞))M1+kxk∞kyk∞M1Zt0+ω t0 η(s, kxk∞,kyk∞)ds. (3.12) After dividing by kxk∞and using the fact that kxk∞,kyk∞<k(x, y)k=r, it gives (3.13) 1 < a(1 −g(r))M1+kyk∞M1Zω 0 η(s, r, r)ds. Similarly, from N2(x, y) = λy, we obtain (3.14) 1 <kxk∞cM2Zω 0 η(s, r, r)ds. Now, adding (3.13) and (3.14) yields 2< a(1 −g(r))M1+M3(kxk∞+kyk∞)Zω 0 η(s, r, r)ds, which in virtue of kxk∞+kyk∞=rcontradicts our assumption (3.7). Therefore, condition (3.10) is satisfied. Now, we prove condition (E2), which reads as follows (3.15) (x, y)−(N1(x, y), N2(x, y)) 6=λ(x0, y0) for every (x, y)∈C, k(x, y)k=R, λ > 0. To this aim, we distinguish again three cases: (a) If x≡0 and y6≡ 0, then N1(x, y), N2(x, y)≡0, so condition (3.15) trivially holds. (b) If x6≡ 0 and y≡0, then N2(x, y)≡0. Therefore condition (3.15) is satisfied since 0< λ y0. (c) Finally, it remains to consider the case when x, y 6≡ 0. If condition (3.15) does not hold, then there exists a pair (x, y)∈Cwith k(x, y)k=Rsuch that x(t)> N1(x, y)(t), y(t)> N2(x, y)(t) for every t∈[0, ω], where we have used that λ, x0, y0>0. 8 CRISTINA LOIS-PRADOS AND RADU PRECUP On the one hand, for each t∈[0, ω], using condition (ii) over ϕ, one has kxk∞≥x(t)> N1(x, y)(t) =Zt+ω t H1(t, s)[a(s)x(s)(1 −g(x(s))) + ϕ(s, x(s), y(s))x(s)y(s)]ds ≥Zt+ω t H1(t, s)ϕ(s, x(s), y(s)) x(s)y(s)ds ≥m1q1q2kxk∞kyk∞Zω 0 ψ(s, x(s) + y(s))ds ≥m1q1q2kxk∞kyk∞Zω 0 ψ(s, k(x, y)k)ds, which after dividing by kxk∞yields (3.16) 1 > m1q1q2kyk∞Zω 0 ψ(s, R)ds. On the other hand, in a similar way, from y(t)> N2(x, y)(t) for every t∈[0, ω], we deduce that (3.17) 1 > m2c q1q2kxk∞Zω 0 ψ(s, R)ds. Now, by adding inequalities (3.16), (3.17) and using kxk∞+kyk∞=k(x, y)k=R, we obtain 2> m3RZω 0 ψ(s, R)ds, which contradicts our assumption (3.8). Thus, condition (3.15) is fulfilled. Therefore, all the conditions of Theorem 2.1 being satisfied, the operator Nhas a fixed point (x, y)∈Cwith r≤ k(x, y)k ≤ R. This fixed point (x, y) is an ω-periodic solution of the Lotka-Volterra type system (1.1).  Remark 3.1. There exists a number r > 0such that conditions (3.6) and (3.7) hold, if (3.18) g(0) >1−1 M1a. Indeed, by using the continuity of gat 0, if (3.18) is satisfied, then there exists r0>0such that g(r)≥1−1/M1a, or equivalently condition (3.6) holds for every r∈(0, r0). From (3.18), we also have g(0) >1−2/M1a, or equivalently (1 −g(0))M1a < 2, which guarantees (3.7) for any small enough r > 0. Notice that condition (3.18) is trivially satisfied when g(0) = 1, which is the case of both linear and logistic growth of the prey population. We consider now the particular expression of gwhich correspond to the linear or logistic growth of the prey population, and we show how the conditions over r, R > 0 in Theorem 3.1 look. Moreover, we study the existence of such numbers rand R, when ϕ≡ϕIor ϕ≡ϕII . For that purpose, let us start by proving that ϕIand ϕII satisfy all the conditions previously required to a general ϕ. It is clear that both functions belong to C(R×R+×R+,R+) LOTKA–VOLTERRA TYPE SYSTEMS 9 and are ω-periodic in the first variable. Concerning conditions (i) and (ii), for function ϕI, we can take η=ϕIand ψ≡λ, while for function ϕII , we can set η=ϕIand ψ(t, z) = λ(t) 1 + β(t)(λ(t) + α(t)z)z. Additionally, we fix the following notations λ:= min s∈[0,ω]λ(s), λ := max s∈[0,ω]λ(s), α := max s∈[0,ω]α(s) and β:= max s∈[0,ω]β(s). 3.2.1. Linear growth. Corollary 3.1. Assume that g≡1and conditions (i),(ii) over ϕare satisfied. If there exist r, R ∈R,0< r < R such that (3.19) rZω 0 η(s, r, r)ds ≤2 M3 , and (3.8) hold, then the system (3.20) (x0=a(t)x−ϕ(t, x, y)xy y0=−b(t)y+c(t)ϕ(t, x, y)xy has an ω-periodic solution (x, y)∈Csuch that r≤ k(x, y)k=kxk∞+kyk∞≤R. Next, we give sufficient conditions for (3.19) and (3.8) to hold, for each one of the two particular expressions of ϕgiven in the Introduction. Case I: When ϕ=ϕI, conditions (3.19), (3.8) read as (3.21) rZω 0 (λ(s) + α(s)r)ds ≤2 M3 , R ≥2 m3Rω 0λ(s)ds and are respectively satisfied provided that (3.22) rλ+αr≤2 M3ω, R ≥2 m3ω λ. Therefore, under condition (3.22), which is satisfied for small enough rand sufficiently large R, Corollary 3.1 applies. Note that the existence of small enough r > 0 is proved also at the end of Remark 3.1. However, the expression in (3.22) tells us how to choose suitable values of rand R. Remark 3.2. We mention that in the particular case of α≡0, system (3.20) turns into (x0=a(t)x−λ(t)xy y0=−b(t)y+c(t)λ(t)xy, which was studied in [13] by means of index theory. Even in this particular case, our result based on Theorem 2.1 gives a better localization of ω-periodic solutions, namely in the annular conical set Cr,R := {(x, y)∈C:r≤ kxk∞+kyk∞≤R}, 16 CRISTINA LOIS-PRADOS AND RADU PRECUP Corollary 3.4. Assume that g(x) = (1 −x/K)and ϕ,η,ψfulfill conditions (i),(ii) without depending on t∈R. If there exist r, R ∈R,0< r < R such that r≤K a M1 ,M1a Kr+M3r η(r, r)≤2, R ψ(R)≥2 m3 , then system (3.29) has an ω-periodic solution (x, y)∈Csuch that r≤ k(x, y)k=kxk∞+kyk∞≤R. 3.6. Case of constant coefficients. Under the conditions required to ϕin Subsection 3.5, we now assume that a,band care constant. Then H1(t, s) = e−a(s−t) 1−e−a ω , H2(t, s) = eb(s−t) eb ω −1. Moreover, for every t∈R, one can compute M1=m1=Zt+ω t H1(t, s)ds =1 a, M2=m2=Zt+ω t H2(t, s)ds =1 b, m3=q1q2min 1 a,c b, M3= max 1 a,c b. As a direct consequence of Theorem 3.4, we have the following result, where the conditions over r, R look much more simpler. Corollary 3.5. Let a,b,cbe constant and ϕ,η,ψin conditions (i),(ii) do not depend on time. If there exist r, R ∈R,0< r < R such that (3.40) g(r)≥0, rη(r, r) max 1 a,c b−g(r)≤1, R ψ(R)≥2 q1q2min 1 a,c b, then the system (3.41) (x0=axg(x)−ϕ(x, y)xy y0=−by +cϕ(x, y)xy, has an ω-periodic solution (x, y)∈Csuch that r≤ k(x, y)k=kxk∞+kyk∞≤R. Notice that, in particular if g≡1 (linear growth on prey), condition (3.40) becomes rη(r, r)≤2 max 1 a,c b and we have the following remark about the classical Lotka-Volterra system: LOTKA–VOLTERRA TYPE SYSTEMS 17 Remark 3.5. In the particular case where ϕ≡λ > 0, system (3.41) reduces to the classical Lotka-Volterra model (1.2). Then, one can consider η, ψ =ϕ=λ, and the conditions over rand Rreduce to r≤2 max 1 a,c bλ= 2 min a, b c1 λ, R≥2 q1q2min 1 a,c bλ= 2 max a, b c1 q1q2λ. It is easy to see that the non-trivial steady state (x∗, y∗) := (b/ (cλ), a/λ)satisfies r≤2 min a, b c1 λ≤ k(x∗, y∗)k=b c+a1 λ≤2 max a, b c1 λ ≤2 max a, b c1 q1q2λ≤R. Therefore, it may happen that the localized solution given by Corollary 3.5 is in fact the steady state (x∗, y∗). In case that g(x) = (1 −x/K) (logistic growth on prey), condition (3.40) becomes r≤K, max 1 a,c br η(r, r) + r K≤2 and we can make the following remark about the classical Lotka-Volterra model with logistic growth of the prey population: Remark 3.6. In the particular case where ϕ≡λ > 0, system (3.41) reduces to the classical Lotka-Volterra model with logistic growth on prey (x0=ax 1−x K−λxy y0=−by +cλxy. Then, one can consider η, ψ =ϕ=λ, and the conditions over rand Rbecome r≤K, r ≤2 1 K+ max 1 a,c bλ, (3.42) R≥2 max a, b c1 λ. If one has K > 2 max a, b c1 λ, then (3.42) holds for R= 2 max {a, b/c}/λ < K. Therefore, we are not localizing the steady state (K, 0), but it may also happen that the localized solution given by Corollary 3.5 is in fact the steady state (x∗, y∗) = b c λ,a λ1−b c λ K , where x∗, y∗>0. 18 CRISTINA LOIS-PRADOS AND RADU PRECUP Acknowledgements The research of Cristina Lois-Prados has been partially supported by grant MTM201675140-P (AEI/FEDER, UE) and grant ED481A-2018/080 from Xunta de Galicia. References [1] H. Baca¨er. A short history of mathematical population dynamics: Lotka-Volterra and the predator-prey system (1920-1926). London: Springer, 2011, pp. 71-76. [2] L. Berec. Impacts of foraging facilitation among predators on predator-prey dynamics. Bull. Math. Biol., 72 (2010), 94-121. [3] F. Brauer and C. Castillo-Ch´avez. Mathematical Models in Population Biology and Epidemiology. New York: Springer, 2001. [4] G. Buffoni, M. Groppi and C. Soresina. Effects of prey over-undercrowding in predator-prey systems with prey-dependent trophic function. 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