On the existence of dead cores for degenerate Lotka-Volterra models

ON THE EXISTENCE OF DEAD CORES FOR DEGENERATE LOTKA-VOLTERRA MODELS Manuel DELGADO and Antonio SU´ AREZ Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, C/ Tarfia s/n, Universidad de Sevilla, 41012-Sevilla, Spain. e-mail: [email protected] and [email protected] Abstract In this work we study the existence, uniqueness and qualitative properties of nonnegative solutions of the Lotka-Volterra models with nonlinear diffusion under homogeneous Dirichlet boundary conditions. We consider the three typical interactions: prey-predator, competition and symbiosis. Unlike the linear diffusion models, nontrivial nonnegative solutions can exist which are not strictly positive. Sufficient conditions in terms of the coefficients involved in the setting of the models are given assuring that one species (or both) does not survive on a set of its habitat (called “dead core”) of positive measure. AMS Classification Codes: 35B50, 35B99, 35J55, 35K57, 92D25. Keywords and phrases: degenerate Lotka-Volterra models, coexistence states, dead core set. 1 1 Introduction In this paper we are interested in nonnegative solutions of the following system      −d1∆Um=U(A−BU ±CV ) in Ω, −d2∆Vn=V(D−EV ±FU) in Ω, U=V= 0 on ∂Ω, (1) where Ω is a bounded domain of IRN,N≥1, with a smooth boundary ∂Ω and m, n, d1, d2, B, C,E,Fare positive constants with m, n ≥1 and A, D ∈IR. The problem (1) models the interactions between two species, with population densities U(x) and V(x), inhabiting the region Ω. Moreover, we are assuming that Ω is fully surrounded by inhospitable areas, because both population densities are subject to homogeneous Dirichlet boundary conditions. Nonlinear diffusion arises mainly in the porous media equation and it was introduced in population dynamics in [16]. We refer to [12] and the references therein for details about the model. For the present model Uand Vcan interact in three different ways indicated by the sign of the last terms in the equations: if both of them are negative then Uand Vcompete; if they are positive then U and Vco-operate; and if, for example, the sign is positive in the first equation and negative in the second one then Uand Vrepresent the predator and prey populations respectively. To study (1), we make an appropiate change of variables (see [12]) and obtain      −∆wm=w(λ−w±bz) in Ω, −∆zn=z(µ−z±cw) in Ω, w=z= 0 on ∂Ω, (2) where λ, µ ∈IR and b, c > 0. When m=n= 1, (2) is the classical Lotka-Volterra model which has been widely studied in the last years: see [4], [6], [8], [9], [15], [23], [26], [28] in competition, [4], [9], [10], [21], [25] in predator-prey and [11], [19], [22], [24], [27] in symbiosis, for instance. When m, n > 1 there exists an important change in the behaviour of the solutions of (2). More precisely, it is possible that one species (or both) is nonnegative but not strictly positive, i.e. there exist some subsets of Ω with strictly positive measures where the species does not survive. We call them “dead cores” (see [14]). For that, we distinguish two types of nontrivial nonnegative solutions of (2); those with both components positive, the coexistence states, and those where at least one component could have a dead core, the semi-coexistence states. We now describe the distribution and the contents of this work. In Section 2 we study the single boundary value problem (−∆wm=w(a(x)−dw) in Ω, w= 0 on ∂Ω, (3) which appears when one of the species is zero and where a∈Cα(Ω), α∈(0,1), m≥1 and d≥0. If m= 1 it is known that (3) admits a unique positive solution if and only if σΩ 1(−a)<0, where σΩ 1(q) stands for the principal eigenvalue of the problem (−∆w+q(x)w=σw in Ω, w= 0 on ∂Ω, 2 with q∈L∞(Ω). The particular case m > 1 and d= 0 has been studied in [1], [2], [3] and [17]. Thus, we focus on the case m > 1 and d > 0 which has been previously analyzed in [17] and [30]. To state our main results, we need the following notation. For any a∈Cα(Ω) we denote aL:= min x∈Ω a(x)aM:= max x∈Ω a(x). In Theorem 1 we obtain some results concerning existence and uniqueness of nonnegative solutions of (3) which can be summarized as follows: •If aL≥0 and 1 < m ≤2 then there exists a unique positive solution of (3). •If aM>0 then there exists at least a nonnegative solution of (3). •If aM≤0 then (3) does not admit a nonnegative solution. Particular attention is paid to the existence of a dead core for solutions of (3). We show: •If either aL>0 or aL= 0 and 1 < m ≤2, any solution of (3) is positive and so it has not a dead core. •If aL= 0, m > 2 and A0:= int{x∈Ω : a(x)=0} 6=∅, there exists a positive constant C=C(N, d, ε) such that if aM< C then any nonnegative solution of (3) has a dead core, where ε > 0 is the radius of a ball contained in A0. When achanges sign, we write at(x) := a+(x)−ta−(x) where a±(x) := max{0,±a(x)}and for any t > 0 and R > 0 we consider the sets N(R, t) := {x∈Ω : a−(x)≥R t}, M(R, t) := {x∈N(R, t) : dist(x, ∂N(R, t)\∂Ω) ≥CR}, with CR:= sN 2Z(a+ M d)m 0 ds [Zs 0 (Rν1/m +dν2/m)dν]1/2. With this notation, we obtain that if M(R, t)6=∅for some t > 0 and R > 0 then any nonnegative solution of (3) has a dead core. Moreover, we consider tas parameter and we obtain information about the qualitative behaviour of the nonnegative solutions of (3). Our results in this direction can be stated in the following general terms: •There exists t0>0 such that if t≥t0then any nonnegative solution of (3) has a dead core. •If 1 < m ≤2 there exists t1>0 such that if 0 < t ≤t1then any nonnegative solution of (3) is positive. •If m > 2 there exist d0>0 and t0(d0)>0 such that if 0 < t ≤t0(d0) then any nonnegative solution of (3) has a dead core. 3 These results improve and extend those in [30] (See Remark 5 and Section 2). In Section 3 we analyze the system (2) in the three classical interactions: predator-prey, competition and symbiosis. The prey-predator case has been investigated in [7], [18], [20], [29], [31]; competition one in [7], [29]; and lastly, the symbiosis one in [7]. We consider λand µas principal parameters and we study the set of values of (λ, µ) for which (2) admits a semi-coexistence or coexistence state. On the other hand, we obtain a priori bounds for nonnegative solutions of (2) which are the key to get the non-existence regions of nonnegative solutions in the (λ, µ)-plane. We should notice that our results improve in some sense those in the above mentioned works (see Remarks 8, 9 and 11). Now, we are interested in the existence of a dead core for any nonnegative solution of (2). This has been previously studied in the cases of prey-predator and competition in [18] and [29]. To get results concerning this subject, we apply the results of Section 2 and obtain for the prey-predator case: •For fixed λ > 0, there exists µ0(λ)>0 such that if 0 < µ < µ0(λ) any nonnegative solution of (2) has a dead core. •For fixed µ > 0, there exist λ0(µ)<0 and ε > 0 such that if λ≤λ0(µ) there is no nonnegative solution of (2) and if λ∈(λ0(µ), λ0(µ) + ε) any nonnegative solution of (2) has a dead core. •For fixed λ, µ, b > 0, there exists c0>0 such that if c > c0any nonnegative solution of (2) has a dead core. In the case of competition, we have: •For fixed λ > 0 (resp. µ > 0), there exists µ0(λ)>0 (resp. λ0(µ)>0) such that if 0< µ < µ0(λ) (resp. 0 < λ < λ0(µ)) any nonnegative solution of (2) has a dead core. •For fixed λ, µ > 0 and c > 0 (resp. b > 0), there exists b0>0 (resp. c0>0) such that if b > b0(resp. c > c0) any nonnegative solution of (2) has a dead core. Moreover, in this Section we will give a biological interpretation to these results. 2 The logistic equation with nonlinear diffusion In this paper we use the following notation: Ω is a bounded domain in IRNwith a smooth enough boundary ∂Ω. For fixed α > 0, we consider the spaces U:= {w∈C2,α(Ω) : w= 0 on ∂Ω} and V:= Cα(Ω) ordered by their cones of nonnegative functions PU:= {w∈ U :w≥0}and PV:= {w∈ V :w≥0}. We will write f≥gif f−g∈P,f > g if f−g∈P−{0}and fÀg if f−g∈˙ P, where ˙ Pdenotes the interior of P. Moreover, for any f∈ V we denote fM:= max x∈Ω f(x)fL:= min x∈Ω f(x). 4 Let a∈ V,m > 1 and d > 0, we consider the logistic equation with nonlinear diffusion, namely (−∆wm=a(x)w−dw2in Ω, w= 0 on ∂Ω. (4) Given q∈L∞(Ω), σΩ 1(q) stands for the first eigenvalue of (−∆w+q(x)w=σw in Ω, w= 0 on ∂Ω, (5) whose corresponding associated eigenfunction ϕΩ 1[q] can be chosen such that ϕΩ 1[q]À0 and kϕΩ 1[q]k∞= 1. Moreover, due to the strong maximun principle, we have ∂ϕΩ 1[q] ∂n <0 (6) where nis the outward unit normal at ∂Ω. Finally, we write σΩ 1:= σΩ 1(0) and ϕΩ 1:= ϕΩ 1[0]. To study (4), we perform the change of variables wm=uand we obtain (−∆u=a(x)u1/m −du2/m in Ω, u= 0 on ∂Ω. (7) For the existence and uniqueness of nonnegative solutions of (7), the main result is the following one: Theorem 1 Let a∈ V,d > 0and m > 1. The following assertions are true: 1. If aÀ0, then there exists a solution u∈ U of (7) with uÀ0. Moreover, if m≤2the solution is unique. 2. If a > 0and m≤2, then there exists a unique solution u∈ U of (7) with uÀ0. 3. If there exists x0∈Ωsuch that a(x0)>0, then there exists at least a solution u∈ U of (7) with u > 0. Remark 1 If a≤0then u= 0 is the unique nonnegative solution of (7) owing to the maximum principle. On the other hand, observe that the function acan change of sign in case 3. Proof. We will apply the sub-supersolution method. By the regularity of athere exist x0∈Ω and r > 0 such that a(x)≥a0>0,for x∈B(x0, r), where B(x0, r) is the ball of radius r > 0 centered at x0. Now, we define the function φ(x) := (ϕΩ0 1(x) if x∈Ω0, 0 if x∈Ω\Ω0, 5 with Ω0= Ω in the case 1 and Ω0=B(x0, r) in the other cases. We will show that u=ρφ is a subsolution of (7) with ρa positive constant. Indeed, we can take ρ > 0 such that −∆(ρϕΩ0 1)≤a(x)(ρϕΩ0 1)1/m −d(ρϕΩ0 1)2/m in Ω0. By (6) we can use Lemma I.1 of [3] and conclude that uis a subsolution of (7). As a supersolution we pick u=M, with M > 0 a sufficiently large constant. Moreover, it is easy to see that kuk∞≤M:= µaM d¶m (8) for any solution uof (7). So, there exists at least a solution u∈ U such that u≤u≤u. Clearly uÀ0 in the first case. Now, we show that uÀ0 in the case 2 by applying the strong maximum principle. Let u > 0 be a solution of (7); it suffices to find a constant K > 0 such that (−∆ + K)u > 0, or equivalently K > du2/m−1−a(x)u1/m−1,with u∈(0, M],(9) which follows from hypotheses and so uÀ0. For the uniqueness we use [5]. We define the map t7→ g(x, t) := a(x)t1/m −dt2/m t=a(x)t1/m−1−dt2/m−1. It is not hard to prove that this application is decreasing in the cases aÀ0 and m≤2 as well as a > 0 and m < 2. It remains to prove the uniqueness in the case m= 2 and a > 0. Let u1and u2be two arbitrary solutions of (7). Now, the function gis nonincreasing, so by the Remark 1 of [5], it follows that there exists a constant Csuch that u1=Cu2. Hence, a(x)u1/2 1−du1=−∆u1=−∆(Cu2) = C(a(x)u1/2 2−du2) = Ca(x)u1/2 2−du1 from which a(x)C1/2u1/2 2(1 −C1/2) = 0, and therefore C= 1. This completes the proof. • Remark 2 1.- When aL>0there exists a unique positive solution of (7) provided that m > 2and aL aM >m−2 m−1. Indeed, it can be proved that the function t7→ g(x, t)defined in the proof of Theorem 1 is decreasing. 2.- In the particular case a(x)≡λ=cte it is known, cf. [13], that there exists a unique positive solution of (7) if and only if λ > 0. 3.- When aÀ0and m > 1, any solution of (7) is positive in Ω. In this case, it is easy to prove the existence of the constant Ksatisfying (9). 6 ¿From the priori bound (8) follows the existence of a nontrivial maximal solution of (7) which we denote by θ[a,d,m]. We will take θ[a,d,m]= 0 if a≤0. The next result will be very useful to compare positive solutions of different logistic boundary value problems. Proposition 1 Let a, b ∈ V. 1. If uis a subsolution of (7), then u≤θ[a,d,m]. 2. If a≤b, then θ[a,d,m]≤θ[b,d,m]. Proof. Let ube a subsolution of (7). As sufficiently large constants, say K > 0, are supersolutions, then there exists a solution uof (7) such that u≤u≤K. From the maximality of θ[a,d,m] it follows that u≤θ[a,d,m]. For the second part it is enough to prove that θ[a,d,m]is a subsolution of (−∆w=b(x)w1/m −dw2/m in Ω, w= 0 on ∂Ω, and apply the previous result. • 2.1 Existence of a “dead core” In this subsection we shall consider the existence of a “dead core” for solutions uof (7), i.e., we will show that the set Ω0:= {x∈Ω : u(x) = 0}has a strictly positive measure under suitable easily checked hypotheses on a,d,m, Ω and N. Assume that a > 0 and A0:= int{x∈Ω : a(x) = 0} 6=∅. Theorem 1 ensures that if m≤2 any nonnegative solution of (7) is positive. Our principal result is: Theorem 2 Let m > 2,a > 0and A0=int{x∈Ω : a(x)=0} 6=∅. Let x0∈A0and ε > 0be such that B(x0,2ε)⊂A0and assume that aM<dm−1 m−2ε2 m−2 [q(q−1 + N−1 2)] 1 m−2 ,(10) with q= 2m/(m−2). Then there exists a dead core for any u > 0solution of (7). Moreover, we have Ω0={x∈Ω : u(x) = 0} ⊃ B(x0, ε). 7 Proof. Let x0∈A0and ε > 0 be such that B(x0,2ε)⊂A0. We will build a function which is zero in B(x0, ε)⊂Ω and bigger than the maximal solution of (7). Without loss of generality we can suppose that x0= 0. We consider the function Ψ(x) :=      0 if x∈B(0, ε), (|x|−ε)q εqif x∈B(0,2ε)\B(0, ε), 1 if x∈Ω\B(0,2ε). It is clear that Ψ ∈H1(Ω). We will show that the function u0:= KΨ satisfies θ[a,d,m]≤u0in Ω, with an appropiate constant K. Taking K≥(aM d)m(11) we have u0≥θ[a,d,m]in Ω\B(0,2ε) from (8). We will prove now that (θ[a,d,m]−u0)+= 0 in B(0,2ε). It is sufficient to prove that ZB(0,2ε)|∇(θ[a,d,m]−u0)+|2≤0,(12) because (θ[a,d,m]−u0)+= 0 on ∂B(0,2ε). Let φ∈H1 0(B(0,2ε)), φ ≥0. Then ZB(0,2ε)∇(θ[a,d,m]−u0)·∇φ=ZB(0,2ε)∇θ[a,d,m]·∇φ−KZB(0,2ε)\B(0,ε)∇Ψ·∇φ= ZB(0,2ε) θ1/m [a,d,m](a(x)−dθ1/m [a,d,m])φ+KZB(0,2ε)\B(0,ε) ∆Ψ ·φ−KZ∂B(0,ε) ∂Ψ ∂n φ. Now using that ∂Ψ/∂n = 0 on ∂B(0, ε) and a(x) = 0 in B(0,2ε), we find that ZB(0,2ε)∇(θ[a,d,m]−u0)·∇φ≤ −ZB(0,2ε) dθ2/m [a,d,m]φ+KZB(0,2ε)\B(0,ε) ∆Ψ ·φ= =−ZB(0,ε) dθ2/m [a,d,m]φ+ZB(0,2ε)\B(0,ε) (−dθ2/m [a,d,m]+K(q εq(|x|−ε)q−2(q−1 + N−1 |x|(|x|−ε))))φ. Taking φ= (θ[a,d,m]−u0)+, we obtain ZB(0,2ε)|∇(θ[a,d,m]−u0)+|2≤ ≤ZB(0,2ε)\B(0,ε) (−dθ2/m [a,d,m]+K(q εq(|x|−ε)q−2(q−1 + N−1 |x|(|x|−ε))))(θ[a,d,m]−u0)+. (13) Using (13) and noting that q−2 = 2q/m, to prove (12) it is sufficient to show that K1−2/m q ε2(q−1 + N−1 |x|(|x|−ε)) −d≤0x∈B(0,2ε)\B(0, ε). 8 We denote K1:= K1−2/m ε2q(q−1) −dand K2:= K1−2/m ε2q(N−1). We must prove that K1+K2≤εK2 |x|for ε < |x|<2ε. (14) Assume that N≥2. If K1+K2≤0, (14) is trivial because K2>0. If K1+K2>0, (14) is equivalent to |x| ≤ εK2 K1+K2 . This inequality is true in B(0,2ε)\B(0, ε) if 2K1+K2≤0.(15) If N= 1 then K2= 0, and so (15) is equivalent to be K1≤0. It is not hard to prove from (10) the existence of a constant Ksatisfying (11) and (15). This finishes the proof. • Remark 3 We remark that the constant C:= C(d, ε, m, N) = dm−1 m−2ε2 m−2 [q(q−1 + N−1 2)] 1 m−2 satisfies C↑+∞as d↑+∞or ε↑+∞. Then, if a > 0is given, Ω0exists if dis large enough. On the other hand, if dis given, Ω0exists if aMis sufficiently small or if A0is large. Now we consider the case where achanges sign. Let t∈IR+and we define at(x) := a+(x)−ta−(x) where a±(x) := max{0,±a(x)}and we suppose that a+6≡ 0. The aim is now to study the existence of a dead core of solutions of (−∆u=at(x)u1/m −du2/m in Ω, u= 0 on ∂Ω, (16) with tas a parameter. To prove the main result we need the following Lemma motivated by Lemma 7 in [30]. Lemma 1 Let R > 0, R0:= sN 2Z+∞ 0 ds [Zs 0 (Rν1/m +dν2/m)dν]1/2 9 Theorem 6 The following assertions are true: 1. Assume λ > 0. Then there exists ρ(λ)>0such that if 0<µ<ρ(λ)any nonnegative solution of (20) has a dead core. 2. Assume µ > 0. Then there exists λ0(µ)<0and ε > 0such that if λ∈(λ0(µ), λ0(µ) + ε) any nonnegative solution of (20) has a dead core. 3. Assume µ > 0and λ > 0. Then there exists c0>0such that if c > c0any nonnegative solution of (20) has a dead core. Proof. Using Propositions 1 and 2 it is easy to prove that for any nonnegative solution (u, v) of (20) we have u≤θ[λ+bθ1/n [µ,n],m]v≤θ[µ−cθ1/m [λ,m],n],(26) and so, Ωu,0:= {x∈Ω : u(x) = 0} ⊃ {x∈Ω : θ[λ+bθ1/n [µ,n],m](x) = 0}, Ωv,0:= {x∈Ω : v(x) = 0} ⊃ {x∈Ω : θ[µ−cθ1/m [λ,m],n](x) = 0}. To prove the first part we will use Theorem 3 and Remark 5. In this case a(x) = µ−cθ1/m [λ,m](x). Hence aM=µand A−={x∈Ω : µ−cθ1/m [λ,m](x)<0}.Let x0∈Ω and R > 0 be such that θ1/m [λ,m](x0) = (θ1/m [λ,m])Mand c(θ1/m [λ,m])M−R > 0. We define B(x) := cθ1/m [λ,m](x)−R, and so, N(R, 1) := N(µ) = {x∈Ω : µ≤B(x)}. It is easily seen that if µ1≤µ2then N(µ2)⊂N(µ1).(27) Let δ > 0 be such that 0 < δ < c(θ1/m [λ,m])M−Rand define µ0:= c(θ1/m [λ,m])M−R−δ. With these choices we can see that x0∈N(µ0) and from the continuity of B(x) there exists rµ0>0 such that B(x0, rµ0)⊂N(µ0). Moreover, from (27) we get B(x0, rµ0)⊂N(µ0)⊂N(µ) for µ∈[0, µ0]. On the other hand, if µ↓0 then aM=µ↓0, and so CR↓0. Thus, there exists µ1>0 such that if 0 < µ < µ1then CR< rµ0. Finally, we take ρ:= min{µ0, µ1}. Thus, for µ∈(0, ρ) CR< rµ0and B(x0, rµ0)⊂N(µ) 16 and so, dist(x0, ∂N(µ)\∂Ω) ≥dist(x0, ∂B(x0, rµ0)\∂Ω) = rµ0> CR. Therefore M(R, 1) 6=∅,and so the first part follows from Theorem 3. In order to prove the second part let a(x) = λ+bθ1/n [µ,n](x) and so, aM=λ+b(θ1/n [µ,n])M. We take λ0=−b(θ1/n [µ,n])M. It is clear that if λ > λ0then aM>0. Let Rand δbe such that 0 < R < δ < −λ0. Thus, for λ∈(λ0,−δ] the set N(R, 1) = {x∈Ω : λ≤ −R−bθ1/n [µ,n](x)} 6=∅. Observe that if λ↓λ+ 0then aM↓0, and so CR↓0. Now, reasoning as in the above paragraph, the result follows. To prove the last part we fix λ, µ > 0. We take R=c1/2and denote by N(R, 1) := N(c). Thus, N(c) = {x∈Ω : µ+c1/2 c≤θ1/m [λ,m](x)}. Observe that the function f(c) := µ+c1/2 c is decreasing and tends to 0 as c↑ ∞. Moreover, if c1≤c2then N(c1)⊂N(c2).Hence, for c sufficiently large we get N(c)6=∅. Finally, observe that CR(c)↓0 as c↑ ∞. We can complete the proof by using an argument similar to that used previously. • Remark 7 We can give a biological interpretation of the above result. In the part 1 we fix the predator’s growth rate. Then if the prey’s growth rate is small, the prey do not live in all the space Ω, i.e., there exist some subregions of Ωwhere the prey is driven to extinction by the predator. On the other hand, in part 3 if the rate at which the prey is consumed by the predator (the parameter cin the setting of the system (20)) is sufficiently large, then there exist some areas where the prey do not exist. In part 2, from (26) observe that if λ≤ −b(θ1/n [µ,n])M:= λ0then there are no semi-coexistence states of (20). Theorem 6ensures that (20) does not admit a coexistence state if λ∈(λ0, λ0+ε). Remark 8 Our results are improvement on previous results papers. Indeed, in [20] the authors obtained under condition (25) the existence of a coexistence state of (20) but only in the particular case 1< m =n < 2. In [7] the authors showed the existence of a semi-coexistence state when λ and µsatisfy (25). It is clear that condition (22) is weaker than (25). 17 3.2 Competition Consider the following model whose solutions represent the steady states of dynamical models of competing populations      −∆u=λu1/m −u2/m −bu1/mv1/n in Ω, −∆v=µv1/n −v2/n −cv1/nu1/m in Ω, u=v= 0 on ∂Ω. (28) Concerning the existence of semi-coexistence states, we have the following result: Theorem 7 (28) possesses a semi-coexistence state if and only if λ > 0and µ > 0. Proof. Let (u, v) be a semi-coexistence state of (28). Then the maximum principle ensures that λand µhave to be positive. Assume now λ, µ > 0. It is clear that B1:= {x∈Ω : λ−bθ1/n [µ,n](x)>0} 6=∅, B2:= {x∈Ω : µ−cθ1/m [λ,m](x)>0} 6=∅.(29) We have to find a couple (u, u)−(v, v) of sub-supersolutions of (28), see [12], i.e. u, u, v, v ∈ H2(Ω) ∩L∞(Ω), u≤uand v≤vin Ω and u≤0≤uand v≤0≤von ∂Ω and such that          −∆u≤λu1/m −u2/m −bu1/mv1/n in Ω, −∆u≥λu1/m −u2/m −bu1/mv1/n in Ω, −∆v≤µv1/n −v2/n −cv1/nu1/m in Ω, −∆v≥µv1/n −v2/n −cv1/nu1/m in Ω. (30) We pick (u, u) = (θ[λ−bθ1/n [µ,n],m], θ[λ,m]) (v,v) = (θ[µ−cθ1/m [λ,m],n], θ[µ,n]). Using (29) and Theorem 1, it can be shown that u > 0, v > 0. Moreover, from Proposition 1, u≤uand v≤vin Ω. It is not hard to prove that this couple satisfies the inequalities in (30) and this completes the proof. • The proofs of the following results are similar to the predator-prey ones, so we omit them. Proposition 4 Let (u, v)be a semi-coexistence state of (28). Then u≤θ[λ,m]v≤θ[µ,n].(31) With respect to the existence of coexistence states we have the following result whose proof can be found in [12]. Theorem 8 Assume bc < 1and λ > bµ and µ > cλ. (32) Then there exists at least one coexistence state of (28). 18 Finally, concerning conditions on λand µunder which any coexistence state has not a dead core, we obtain: Proposition 5 Assume bc < 1and that λand µsatisfy (32). Then any nonnegative solution of (28) has not a dead core. The following result gives us conditions which ensure that any nonnegative solution of (28) has a dead core. Theorem 9 The following assertions are true: 1. Assume λ > 0. Then there exists µ(λ)>0such that if 0< µ < µ(λ)any nonnegative solution of (28) has a dead core. 2. Assume µ > 0. Then there exists λ(µ)>0such that if 0< λ < λ(µ)any nonnegative solution of (28) has a dead core. 3. Assume λ, µ > 0and b > 0. Then there exists c0>0such that if c > c0any nonnegative solution of (28) has a dead core. 4. Assume λ, µ > 0and c > 0. Then there exists b0>0such that if b > b0any nonnegative solution of (28) has a dead core. Proof. We will prove the first and third parts. The other ones follow similarly. We claim that for any nonnegative solution (u, v) of (28), we have u≤θ[λ−bθ1/n [µ−cθ1/m [λ,m],n] ,m], v ≤θ[µ−cθ1/m [λ−bθ1/n [µ,n],m] ,n].(33) Indeed, using Proposition 1 it is sufficient to show that uis subsolution of   −∆w= (λ−bθ1/n [µ−cθ1/m [λ,m],n])w1/m −w2/m in Ω, w= 0 on ∂Ω, which is true if θ[µ−cθ1/m [λ,m],n]≤v. ¿From Propositions 4 and 1, it follows that v=θ[µ−cu1/m,n]≥θ[µ−cθ1/m [λ,m],n]. This proves the first relation of (33). The second one follows analogously. So, from (33) we have Ωu,0={x∈Ω : u(x) = 0} ⊃ {x∈Ω : θ[λ−bθ1/n [µ−cθ1/m [λ,m],n] ,m](x) = 0}, Ωv,0={x∈Ω : v(x) = 0} ⊃ {x∈Ω : θ[µ−cθ1/m [λ−bθ1/n [µ,n],m] ,n](x) = 0}. We now prove the first part. For fixed λ > 0, we consider a(x) = µ−cθ1/m [λ−bθ1/n [µ,n],m](x), 19 and so aM=µ. As λ > 0, the set {x∈Ω : λ−bθ1/n [µ,n](x)>0} 6=∅and thus θ[λ−bθ1/n [µ,n],m]>0 for µ≥0. Let a(x, µ) := cθ1/m [λ−bθ1/n [µ,n],m](x). ¿From Proposition 1, if µ1≤µ2then a(x, µ2)≤a(x, µ1).(34) On the other hand, a(x, 0) = cθ1/m [λ,m]À0 and so there exists µ0>0 such that for 0 ≤µ<µ0 we have 0≤µ < (a(x, µ0))M≤(a(x, µ))M. By the continuity of a(x, µ0) there exists R0>0 such that for µ < µ0the set {x∈Ω : µ≤a(x, µ0)−R0} 6=∅. Hence, N(µ) := N(R0,1) = {x∈Ω : µ≤a(x, µ)−R0} ⊃ {x∈Ω : µ≤a(x, µ0)−R0} 6=∅. Finally, from (34) it follows easily that if µ1≤µ2then N(µ2)⊂N(µ1).Now we can reason as in the first part of Theorem 6. We will prove the third part. For fixed λ, µ > 0 and b > 0, we consider the same function a(x) as above. In this case, we have N(R, 1) := N(c) = {x∈Ω : µ+R c≤θ1/m [λ−bθ1/n [µ,n],m](x)}, and again arguing as for the second part of Theorem 6 the proof follows. • Remark 9 1. Theorem 9has a biological interpretation similar to Theorem 6given in Remark 7. 2. The existence of semi-coexistence states of (28) has been studied in [7]. The authors obtained a semi-coexistence state under the condition (32), which is stronger than λ, µ > 0. It seems that the existence of coexistence states of (32) has not been analyzed previously. 3.3 Symbiosis In this case we consider the cooperative system      −∆u=λu1/m −u2/m +bu1/mv1/n in Ω, −∆v=µv1/n −v2/n +cv1/nu1/m in Ω, u=v= 0 on ∂Ω. (35) Our main result about the existence of semi-coexistence states is the following one: 20 Theorem 10 Assume bc < 1and that the sets P1:= {x∈Ω : µ+cθ1/m [λ,m](x)>0} 6=∅P2:= {x∈Ω : λ+bθ1/n [µ,n](x)>0} 6=∅. Then there exists at least one semi-coexistence state of (35). Remark 10 An observation about the way in which the above conditions should be interpreted is appropriate. If for example λ≤0then θ[λ,m]= 0, and so P16=∅is equivalent to µ > 0. Proof. Again we use the sub-supersolution method for systems. In this case a couple (u, u)−(v,v) is sub-supersolution of (35) if, see [12], u, u, v, v ∈H2(Ω) ∩L∞(Ω), u≤uand v≤vin Ω and u≤0≤uand v≤0≤von ∂Ω and such that          −∆u≤λu1/m −u2/m +bu1/mv1/n in Ω, −∆u≥λu1/m −u2/m +bu1/mv1/n in Ω, −∆v≤µv1/n −v2/n +cv1/nu1/m in Ω, −∆v≥µv1/n −v2/n +cv1/nu1/m in Ω. (36) We consider two cases: 1. Assume λ≤0. As we have mentioned above, the condition P16=∅is equivalent to µ > 0. We take (u, u) = (θ[λ+bθ1/n [µ,n],m], M) (v, v) = (θ[µ,n], N), with M, N > 0 constants still to be chosen. Since P26=∅then u > 0. It is easy to show that Mand Nsatisfy (36) provided that λ−M1/m +bN1/n ≤0, µ −N1/n +cM1/m ≤0.(37) Since bc < 1 there exist Mand Nsatisfying (37). Thus the proof of case 1 is complete. 2. Assume λ > 0. In this case, P2= Ω. Now we pick (u, u) = (θ[λ,m], M) (v,v) = (θ[µ+cθ1/m [λ,m],n], N), with Mand Nas above. This completes the proof. • Again the following results do not need any proof. Proposition 6 Assume bc < 1and let (u, v)be a nonnegative solution of (35). Then θ[λ,m]≤u≤µλ+bµ 1−bc ¶m θ[µ,n]≤v≤µµ+cλ 1−bc ¶n .(38) As an immediate consequence of Proposition 6, we have 21 Corollary 5 Assume bc < 1and either λ+bµ ≤0or µ+cλ ≤0. Then (35) does not admit a semi-coexistence state. We obtain the following result on coexistence states Theorem 11 Assume bc < 1and (λ, µ)∈IR+×IR+\{(0,0)}.(39) Then (35) possesses at least one coexistence state. Proof. Assume λ, µ > 0. We take (u, u) = (θ[λ,m], M) (v,v) = (θ[µ,n], N) with Mand Nsatisfying (37). It is not difficult to show that this couple is a sub-supersolution of (35). On the other hand, if for example λ= 0 and µ > 0 we pick (u, u) = (θ[bθ1/n [µ,n],m], M) (v,v) = (θ[µ,n], N). In this case, Mand Nhave to satisfy the following inequalities −M1/m +bN1/n ≤0, µ −N1/n +cM1/m ≤0. We can reason analogously in the remaining case. This concludes the proof. • The following result can be proved similarly to Proposition 3. Proposition 7 Assume (39). Then any nonnegative solution of (35) is positive, i.e., it has not a dead core. The following result says that the set of (λ, µ)∈IR2where (35) has at least a semi-coexistence state is connected. Theorem 12 Assume bc < 1. Let us denote Γ := {(λ, µ)∈IR2: (35) has at least a semi-coexistence state in (λ, µ)}. Then Γis connected. Proof. Let (u0, v0) be a semi-coexistence state in (λ0, µ0). We fix λ0. Let µ∈[µ0,∞) and define (u, u) = (u0, M) (v,v) = (v0, N) with Mand Nsatisfying (37). It is not hard to show that this couple is a sub-supersolution of (35) in (λ0, µ). This completes the proof. • Remark 11 To our knowledge, (35) has been only studied previously in [7]. In this work the authors obtained the existence of a semi-coexistence state under the condition λ > 0and µ > 0. From Theorem 10,(35) possesses a semi-coexistence state even when λor µis negative. Acknowledgements. We are grateful to Professors J. Hern´andez and J. L. G´amez for helpful comments. The authors thank to DGICYT and CICYT of Spain for research support under grant DGICYT PB95-1242 and MAR98-0486, respectively. 22 References [1] D. G. Aronson and L. A. Peletier, Large time behaviour of solutions of the porous medium equation in bounded domains, J. Diff. Eqns. 39, 378-412 (1981). [2] C. Bandle, M. A. Pozio and A. Tesei, The asymptotic behavior of the solutions of degenerate parabolic equations, Trans. A.M.S. 303, 487-501 (1987). [3] H. Berestycki and P. L. Lions, Some applications of the method of sub and supersolutions, Springer Lectures Notes 782, 16-41 (1980). [4] J. Blat and K. J. Brown, Bifurcation of steady-state solutions in predator-prey and competition systems, Proc. R. Soc. Edinb.,97A, 21-34 (1984). [5] H. Brezis and L. Oswald, Remarks on sublinear elliptic equations, Nonl. Anal. T.M.A.,10, 55-64 (1986). [6] R. S. Cantrell and C. Cosner, On the steady-state problem for the Volterra-Lotka competition model with diffusion, Houston J. Math. 13, 337-352 (1987). [7] A. Ca˜ nada and J. L. G´ amez, Elliptic systems with nonlinear diffusion in population dynamics, Diff. Eqns. Dynam. Sys. 3, 189-204 (1995). [8] C. Cosner and A. C. Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion, S.I.A.M. J. Appl. Math. 44, 1112-1132 (1984). [9] E. N. Dancer, On positive solutions of some pairs of differential equations, part I, Trans. A.M.S. 284, 729-743 (1984). [10] E. N. Dancer, On positive solutions of some pairs of differential equations, part II, J. Diff. Eqns. 60, 236-258 (1985). [11] M. Delgado, J. L´ opez-G´ omez and A. Su´ arez, On the symbiotic Lotka-Volterra model with diffusion and transport effects, to appear in J. Diff. Eqns. [12] M. Delgado and A. Su´ arez, Existence of solutions for elliptic systems with H¨older continuous nonlinearities, to appear in Diff. Int. Eqns. [13] M. Delgado and A. Su´ arez, On the structure of the positive solutions of the logistic equation with nonlinear diffusion, to appear. [14] J. I. D´ ıaz and J. Hern´ andez, On the existence of a free boundary for a class of reaction-diffusion systems, S.I.A.M. J. Math. Anal. 15, 670-685 (1984). [15] J. C. Eilbeck, J. Furter and J. L´ opez-G´ omez, Coexistence in the competition model with diffusion, J. Diff Eqns. 107, 96-139 (1994). [16] M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci. 33, 35-49 (1977). [17] J. Hern´ andez, Positive solutions for the logistic equation with unbounded limitations, in “Reaction diffusion systems”, Lecture Notes in Pure and Applied Mathematics 194, 183-197 (1997). [18] J. Hern´ andez, Some free boundary problems for predator-prey systems with nonlinear diffusion, Proc. Sympos. Pure Math. 45, 481-488 (1986). 23 [19] P. Korman and A. Leung, On the existence and uniqueness of positive steady state in the Volterra-Lotka ecological models with diffusion, Appl. Anal. 26, 145-160 (1987). [20] A. Leung and G. Fan, Existence of positive solutions for elliptic systemsdegenerate and nondegenerate ecological models, J. Math. Anal. Appl. 151, 512-531 (1990). [21] L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. A.M.S. 305, 143-166 (1988). [22] L. Li and A. Ghoreishi, On positive solutions of general nonlinear elliptic symbiotic interacting systems, Appl. Anal. 40, 281–295 (1991). [23] L. Li and R. Logan, Positive solutions to general elliptic competition models, Diff. Int. Eqns. 4, 817-834 (1991). [24] J. L´ opez- G´ omez and R. Pardo, Coexistence regions in Lotka-Volterra models with diffusion, Nonl. Anal. T.M.A. 19, 11-28 (1992). [25] J. L´ opez- G´ omez and R. Pardo, Existence and uniqueness of coexistence states for the predatorprey model with diffusion: the scalar case, Diff. Int. Eqns. 6, 1025-1031 (1993). [26] J. L´ opez- G´ omez and J. C. Sabina, Coexistence states and global attractivity for some convective diffusive competing species models, Trans. A.M.S 347, 3797-3833 (1995). [27] Y. Lou, Necessary and sufficient condition for the existence of positive solutions of certain cooperative system, Non. Anal. T.M.A. 26, 1079-1095 (1996). [28] P. J. McKenna and W. Walter, On the Dirichlet problem for elliptic systems, App. Anal. 21, 207-224 (1986). [29] M. A. Pozio and A. Tesei, Degenerate parabolic problems in population dynamics, Japan J. Appl. Math. 2, 351-380 (1985). [30] M. A. Pozio and A. Tesei, Support properties of solutions for a class of degenerate parabolic problems, Comm. Partial Diff. Eqns. 12, 47-75 (1987). solutions of [31] Z. Shenghai, Existence of nonnegative solutions for degenerate ecological models, Appl. Math. JCU. 13B, 8-14 (1998). 24