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Andrews University Digital Commons @ Andrews University Faculty Publications 7-24-2013 Positive Equilibrium Solutions to General Population Model Joon Hyuk Kang Andrews University, kang@andrews.edu Follow this and additional works at: https://digitalcommons.andrews.edu/pubs Part of the Applied Mathematics Commons Recommended Citation Hyuk Kang, Joon, "Positive Equilibrium Solutions to General Population Model" (2013) Faculty Publications 1689 https://digitalcommons.andrews.edu/pubs/1689 This Article is brought to you for free and open access by Digital Commons @ Andrews University It has been accepted for inclusion in Faculty Publications by an authorized administrator of Digital Commons @ Andrews University For more information, please contact repository@andrews.edu International Journal of Pure and Applied Mathematics Volume 85 No 2013, 1009-1019 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v85i6.4 AP ijpam.eu POSITIVE EQUILIBRIUM SOLUTIONS TO GENERAL POPULATION MODEL Joon Hyuk Kang Department of Mathematics Andrews University Berrien Springs, MI 49104, USA Abstract: In this paper, we investigate conditions for the existence of positive solution to the following general elliptic system with various growth conditions: ∆u + u(a + g(u, v)) = in Ω, ∆v + v(d + h(u, v)) = u|∂Ω = v|∂Ω = Our arguments mainly rely on super-sub solutions, maximum principles, spectrum estimates, and some detailed properties for the solution of logistic equations AMS Subject Classification: 35A05, 35A07 Key Words: elliptic system with various growth conditions, existence of positive solution, super-sub solutions, maximum principles, spectrum estimates, logistic equations Introduction One of the prominent subjects of study and analysis in mathematical biology concerns the competition, predator-prey or cooperation of two or more species of animals in the same environment Especially pertinent areas of investigation include the conditions under which the species can coexist, as well as the conReceived: March 3, 2013 c 2013 Academic Publications, Ltd url: www.acadpubl.eu 1010 J.H Kang ditions under which any one of the species becomes extinct, that is, one of the species is excluded by the others (see [1], [2], [3], [4], [6], [7], [11], [12], [13]) In this paper, we focus on the general population model to better understand the interactions between two species Preliminaries Before entering into our primary arguments and results, we must first present a few preliminary items that we later employ throughout the proofs detailed in this paper The following definition and lemmas are established and accepted throughout the literature on our topic Definition 2.1 (Super and sub solutions) Consider ∆u + f (x, u) = in Ω, u|∂Ω = 0, (1) ¯ × R) and Ω is a bounded domain in Rn where f ∈ C α (Ω ¯ satisfying (A) A function u ¯ ∈ C 2,α (Ω) ∆¯ u + f (x, u ¯) ≤ in Ω, u ¯|∂Ω ≥ is called a super solution to (1) ¯ satisfying (B) A function u ∈ C 2,α (Ω) ∆u + f (x, u) ≥ in Ω, u|∂Ω ≤ is called a sub solution to (1) ¯ be, respec¯ × R) and let u Lemma 2.1 Let f (x, ξ) ∈ C α (Ω ¯, u ∈ C 2,α (Ω) ¯ Then ¯(x), x ∈ Ω tively, super and sub solutions to (1) which satisfy u(x) ≤ u ¯ with u(x) ≤ u(x) ≤ u ¯ (1) has a solution u ∈ C 2,α (Ω) ¯(x), x ∈ Ω In our proof, we also employ accepted conclusions concerning the solutions of the following logistic equations Lemma 2.2 (Established in [13]) Consider ∆u + uf (u) = in Ω, u|∂Ω = 0, u > 0, POSITIVE EQUILIBRIUM SOLUTIONS TO 1011 where f is a decreasing C function such that there exists c0 > such that f (u) ≤ for u ≥ c0 and Ω is a bounded domain in Rn If f (0) > λ1 , then the above equation has a unique positive solution, where λ1 is the first eigenvalue of −∆ with homogeneous boundary conditions whose corresponding eigenfunction is denoted by φ1 We denote this unique positive solution as θf The most important property of this positive solution is that θf is increasing as f is increasing We specifically note that for a > λ1 , the unique positive solution of ∆u + u(a − u) = in Ω, u|∂Ω = 0, u > 0, is denoted by ωa ≡ θa−x Hence, θa is increasing as a > is increasing Consider the system ∆u + f (x, u) = in, Ω, u = on ∂Ω, (2) where u = (u1 , , um ) and f = (f1 , , fm ) is quasimonotone increasing, i.e fi (x, u) is increasing in uj for all j = i Lemma 2.3 ([12]) Let wλ be a family of subsolutions(α ≤ λ ≤ β) to (2), increasing in λ such that ∆wλ + f (x, wλ ) ≥ in Ω, wλ = on ∂Ω Assume also u ≥ wα , wλ does not satisfy (2) for any λ, and continuously in λ on ∂Ω Then u ≥ sup wλ ∂wλ ∂n changes Cooperating Species In [12], Korman and Leung established a sufficient and necessary condition for the existence of positive solution to the cooperation system ∆u(x) + u(x)(a − u(x) + cv(x)) = in Ω, (3) ∆v(x) + v(x)(d − v(x) + eu(x)) = u(x)|∂Ω = v(x)|∂Ω = where Ω is a bounded domain in Rn , u(x) and v(x) designate the population densities for the two species The following is their result: 1012 J.H Kang Theorem 3.1 For existence of a positive solution to (3) it is necessary and sufficient that ce < In this section, we develop their result to more general population model: ∆u(x) + u(x)(a + g(u(x), v(x))) = in Ω, (4) ∆v(x) + v(x)(d + h(u(x), v(x))) = u|∂Ω = v|∂Ω = 0, where a and d are positive constants, g, h ∈ C are such that gu < 0, gv > 0, hu > 0, hv < and g(0, 0) = h(0, 0) = The following Theorem proves a necessary condition for the existence of a positive solution to (4) Theorem 3.2 If a > λ1 , d > λ1 , inf(hv ) ≥ −1, and inf(hu ) > 0, then the existence of a positive solution to (4) implies inf(gv ) inf(hu ) + inf(gu ) < Proof Suppose inf(gv ) inf(hu ) + inf(gu ) ≥ Consider a family (uλ , vλ ) = (λφ1 , λ inf(hu )φ1 ) with any λ > Then by the assumption and Mean Value Theorem, = = = ≥ > ∆uλ + uλ [a + g(uλ , vλ )] −λλ1 φ1 + λφ1 [a + g(λφ1 , λ inf(hu )φ1 )] λφ1 [−λ1 + a + g(λφ1 , λ inf(hu )φ1 )] λφ1 [a − λ1 + g(λφ1 , λ inf(hu )φ1 − g(λφ1 , 0) + g(λφ1 , 0) − g(0, 0)] λφ1 [a − λ1 + inf(gv ) inf(hu )λφ1 + inf(gu )λφ1 ] 0, and = = = ≥ > ∆vλ + vλ [d + h(uλ , vλ )] −λ inf(hu )λ1 φ1 + λ inf(hu )φ1 [d + h(λφ1 , λ inf(hu )φ1 )] λ inf(hu )φ1 [−λ1 + d + h(λφ1 , λ inf(hu )φ1 )] λ inf(hu )φ1 [d − λ1 + h(λφ1 , λ inf(hu )φ1 ) − h(λφ1 , 0) + h(λφ1 , 0) −h(0, 0)] λ inf(hu )φ1 [d − λ1 + inf(hv )λ inf(hu )φ1 + inf(hu )λφ1 ] Therefore, (uλ , vλ ) = (λφ1 , λ inf(hu )φ1 ) with any λ > is a family of subsolutions to (4) POSITIVE EQUILIBRIUM SOLUTIONS TO 1013 Furthermore, if (u, v) is a positive solution to (4), then u > λ0 φ1 and v > λ0 inf(hu )φ1 for suffuciently small λ0 > 0, and so by the lemma 2.3, we conclude that u ≥ λφ1 and v ≥ λ inf(hu )φ1 for any λ ≥ λ0 Hence, there is no positive solution to (4) For a sufficient condition for the existence of a positive solution to (4), we need the following Lemma Lemma 3.3 such that If bf > ce, then we can choose arbitrary large M, N > a − bM + cN < 0, d + eu − f N < We now establish a sufficient condition for the existence of a positive solution to (4) Theorem 3.4 If a > λ1 , d > λ1 and sup(gv ) sup(hu ) < sup(gu ) sup(hv ), then (4) has a positive solution Proof Let u = αφ1 , v = βφ1 , where α, β > Then since a > λ1 and d > λ1 , by the Mean Value Theorem, for small enough α, β > 0, = = = ≥ ≥ ∆u + u[a + g(u, v)] −αλ1 φ1 + αφ1 [a + g(αφ1 , βφ1 )] αφ1 [−λ1 + a + g(αφ1 , βφ1 )] αφ1 [−λ1 + a + g(αφ1 , βφ1 ) − g(0, βφ1 ) + g(0, βφ1 ) − g(0, 0)] αφ1 [−λ1 + a + inf(gu )αφ1 + inf(gv )βφ1 ] 0, = = = ≥ ≥ ∆v + v[d + h(u, v)] −βλ1 φ1 + βφ1 [d + h(αφ1 , βφ1 )] βφ1 [−λ1 + d + h(αφ1 , βφ1 )] βφ1 [−λ1 + d + h(αφ1 , βφ1 ) − h(0, βφ1 ) + h(0, βφ1 ) − h(0, 0)] βφ1 [−λ1 + d + inf(hu )αφ1 + inf(hv )βφ1 ] 0, and and so, (u, v) = (αφ1 , βφ1 ) is a subsolution to (4) for sufficiently small α, β > But, for all (u, v), by the Mean Value Theorem again, g(u, v) = g(u, v) − g(u, 0) + g(u, 0) − g(0, 0) ≤ sup(gv )v + sup(gu )u, 1014 and J.H Kang h(u, v) = h(u, v) − h(u, 0) + h(u, 0) − h(0, 0) ≤ sup(hv )v + sup(hu )u, so by the condition and the Lemma 3.3, there are constants M, N > with αφ1 < M, βφ1 < N such that ∆M + M [a + g(M, N )] ≤ M [a + sup(gv )N + sup(gu )M ] < 0, ∆N + N [d + h(M, N )] ≤ N [d + sup(hv )N + sup(hu )M ] < 0, in other words, (M, N ) is a supersolution to (4) We conclude by the Lemma 2.1 that there is a positive solution (u, v) to (4) with αφ1 ≤ u ≤ M, βφ1 ≤ v ≤ N Competing Species In [4], Cosner and Lazer established a sufficient and necessary condition for the existence of positive solution to the competing system ∆u(x) + u(x)(a − bu(x) − cv(x)) = in Ω, (5) ∆v(x) + v(x)(a − f v(x) − eu(x)) = u(x)|∂Ω = v(x)|∂Ω = 0, where Ω is a bounded domain in Rn , a, b, c, e, f > are constants, u(x) and v(x) designate the population densities for the two species The following is their result: Theorem 4.1 In order that there exist positive smooth functions u and v in Ω satisfying (5), it is necessary and sufficient that one of the following three sets of conditions holds (i)a > λ1 , b > e, c < f (ii)a > λ1 , b = e, c = f (iii)a > λ1 , b < e, c > f In this section, we develop their result to more general population model: ∆u(x) + u(x)[a + g(u(x), v(x))] = in Ω, (6) ∆v(x) + v(x)[a + h(u(x), v(x))] = u|∂Ω = v|∂Ω = 0, where a is a positive constant, g, h ∈ C are such that gu < 0, gv < 0, hu < 0, hv < 0, there exist constants c0 > 0, c1 > such that a + g(u, 0) ≤ for u ≥ c0 and a + h(0, v) ≤ for v ≥ c1 , and g(0, 0) = h(0, 0) = POSITIVE EQUILIBRIUM SOLUTIONS TO 1015 The following theorem provides a sufficient condition for the existence of a positive smooth solution to (6) Theorem 4.2 Suppose one of the following three sets of conditions holds (1)a > λ1 , inf(gu ) < inf(hu ), inf(gv ) > inf(hv ) (2)a > λ1 , inf(gu ) = inf(hu ), inf(gv ) = inf(hv ) (3)a > λ1 , inf(gu ) > inf(hu ), inf(gv ) < inf(hv ) Then (6) has a positive smooth solution Proof By the Theorem 4.1, if one of the above three sets of conditions holds, then there is a positive smooth solution (u, v) to ∆u + u[a − (− inf(gu ))u − (− inf(gv ))v] = ∆v + v[a − (− inf(hu ))u − (− inf(hv ))v] = in Ω, u|∂Ω = v|∂Ω = But, by the Mean Value Theorem, = ≥ = = ∆u + u[a + g(u, v)] ∆u + u[a + g(u, v) − g(0, v) + g(0, v) − g(0, 0)] ∆u + u[a + inf(gu )u + inf(gv )v] ∆u + u[a − (− inf(gu ))u − (− inf(gv ))v] 0, = ≥ = = ∆v + v[a + h(u, v)] ∆v + v[a + h(u, v) − h(0, v) + h(0, v) − h(0, 0)] ∆v + v[a + inf(hu )u + inf(hv )v] ∆v + v[a − (− inf(hu ))u − (− inf(hv ))v] and Hence, (u, v) is a subsolution to (6) But by the conditions of g, h, any large positive constant M satisfying u < M, v < M in Ω is a supersolution to (6) Therefore, by the Lemma 2.1, (6) has a positive smooth solution The next theorem establishes a necessary condition for the existence of a positive smooth solution to (6) Theorem 4.3 If (6) has a positive smooth solution, then a > λ1 and one 1016 J.H Kang of the following six sets of conditions holds (1)gu ≡ hu are constants, inf(hv ) ≤ sup(gv ), sup(hv ) ≥ inf(gv ) (2) inf(hu ) = sup(gu ), sup(hu ) > inf(gu ), inf(hv ) ≤ sup(gv ) (3) inf(hu ) > sup(gu ), sup(hu ) > inf(gu ), inf(hv ) < sup(gv ) (4) inf(hu ) < sup(gu ), sup(hu ) = inf(gu ), sup(hv ) ≥ inf(gv ) (5) inf(hu ) < sup(gu ), sup(hu ) < inf(gu ), sup(hv ) > inf(gv ) (6) inf(hu ) < sup(gu ), sup(hu ) > inf(gu ) Proof Suppose (u, v) is a positive smooth solution to (6) By the Mean Value Theorem, there are u ˜, v˜ with ≤ u ˜ ≤ u, ≤ v˜ ≤ v such that g(u, 0) − g(0, 0) = gu (˜ u, 0)u, g(u, v) − g(u, 0) = gv (u, v˜)v Hence, by the Green’s Identity, u, 0)u − gv (u, v˜)v]dx Ω uφ1 [λ1 − a − gu (˜ = Ω uφ1 [λ1 − a + g(0, 0) − g(u, 0) + g(u, 0) − g(u, v)]dx = Ω uφ1 [λ1 − a − g(u, v)]dx = Ω φ1 [−au − ug(u, v)] + uλ1 φ1 dx = Ω φ1 ∆u − u∆φ1 dx = But, since −gu (˜ u, 0)u − gv (u, v˜)v > in Ω, a > λ1 By the Mean Value Theorem again, there are u1 , u2 , v1 , v2 with ≤ u1 , u2 ≤ u, ≤ v1 , v2 ≤ v such that g(u, v) − g(0, v) = gu (u1 , v)u, h(u, v) − h(0, v) = hu (u2 , v)u, g(0, v) − g(0, 0) = gv (0, v1 )v, h(0, v) − h(0, 0) = hv (0, v2 )v Therefore, by the Green’s Identity again, = = Ω uv([hu (u2 , v) − gu (u1 , v)]u + [hv (0, v2 ) − gv (0, v1 )]v)dx Ω uv[hu (u2 , v)u + hv (0, v2 )v − gu (u1 , v)u − gv (0, v1 )v]dx Ω uv[h(u, v) − h(0, v) + h(0, v) − h(0, 0) + g(0, v) − g(u, v) − g(0, v) +g(0, 0)]dx = Ω uv[h(u, v) − g(u, v)]dx = Ω v∆u − u∆vdx = 0, POSITIVE EQUILIBRIUM SOLUTIONS TO 1017 and so, uv([inf(hu ) − sup(gu )]u + [inf(hv ) − sup(gv )]v)dx ≤ 0, Ω uv([sup(hu ) − inf(gu )]u + [sup(hv ) − inf(gv )]v)dx ≥ 0, Ω which derives (A) inf(hu ) = sup(gu ), inf(hv ) ≤ sup(gv ), (B) inf(hu ) > sup(gu ), inf(hv ) < sup(gv ), (C) inf(hu ) < sup(gu ), and (A′ ) sup(hu ) = inf(gu ), sup(hv ) ≥ inf(gv ), (B ′ ) sup(hu ) < inf(gu ), sup(hv ) > inf(gv ), (C ′ ) sup(hu ) > inf(gu ) Combining (A), (B), (C) and (A′ ), (B ′ ), (C ′ ) together, we may have (A”)gu ≡ hu are constants, inf(hv ) ≤ sup(gv ), sup(hv ) ≥ inf(gv ), (B”) inf(hu ) = sup(gu ), sup(hu ) < inf(gu ), inf(hv ) ≤ sup(gv ), sup(hv ) > inf(gv ), (C”) inf(hu ) = sup(gu ), sup(hu ) > inf(gu ), inf(hv ) ≤ sup(gv ), (D”) inf(hu ) > sup(gu ), sup(hu ) = inf(gu ), inf(hv ) < sup(gv ), sup(hv ) ≥ inf(gv ), (E”) inf (hu ) > sup(gu ), sup(hu ) < inf(gu ), inf(hv ) < sup(gv ), sup(hv ) > inf(gv ), (F ”) inf(hu ) > sup(gu ), sup(hu ) > inf(gu ), inf(hv ) < sup(gv ), (G”) inf (hu ) < sup(gu ), sup(hu ) = inf(gu ), sup(hv ) ≥ inf(gv ), (H”) inf(hu ) < sup(gu ), sup(hu ) < inf(gu ), sup(hv ) > inf(gv ), (I”) inf(hu ) < sup(gu ), sup(hu ) > inf(gu ) However, it is clear that (B”), (D”), (E”) are not possible, so we establish the result of the Theorem Finally, we prove a nonexistence result Theorem 4.4 If a > µν d, −1 ≤ gu < 0, and hv ≤ −1, where µ = min[− sup(hu ), 1] and ν = max[− inf(gv ), 1], then there is no positive solution to (6) Proof Suppose there is a positive solution (u, v) to (6) Then by the Mean Value Theorem, the Green’s Identity and the inequality 1018 J.H Kang conditions, ≤ ≤ = = = But, Ω (a − d + [− sup(hu ) − 1]u + [1 + inf(gv )]v)uvdx (a − d + [inf(gu ) − sup(hu )]u + [inf(gv ) − sup(hv )]v)uvdx Ω Ω [a − d + g(u, 0) − g(0, 0) − h(u, v) + h(0, v) + g(u, v) − g(u, 0) − h(0, v) +h(0, 0)]uvdx Ω [a − d + g(u, v) − h(u, v)]uvdx Ω (v∆u − u∆v)dx (7) if a > µν d, then since a ≥ u and d ≥ v, a − d + [− sup(hu ) − 1]u + [1 + inf(gv )]v ≥ µa − νd > 0, which contradicts to (7) References [1] S.W Ali, C Cosner, On the uniqueness of the positive steady state for Lotka-Volterra Models with diffusion, Journal of Mathematical Analysis and Application, 168 (1992), 329-341 [2] R.S Cantrell, C Cosner, On the steady-state problem for the VolterraLotka competition model with diffusion, Houston Journal of Mathematics, 13 (1987), 337-352 [3] R.S Cantrell, C Cosner, On the uniqueness and stability of positive solutions in the Volterra-Lotka competition model with diffusion, Houston J Math., 15 (1989), 341-361 [4] C Cosner, A.C Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion, Siam J Appl Math., 44 (1984), 11121132 [5] D Dunninger, Lecture Note for Applied Analysis in Michigan State University [6] J L.-Gomez, R Pardo, Existence and uniqueness for some competition models with diffusion, C.R Acad Sci Paris, Ser I Math., 313 (1991), 933-938 POSITIVE EQUILIBRIUM SOLUTIONS TO 1019 [7] C Gui, Y Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm Pure and Appl Math., XVL2, No 12 (1994), 1571-1594 [8] J Kang, Y Oh, A sufficient condition for the uniqueness of positive steady state to a reaction diffusion system, Journal of Korean Mathematical Society, 39, No 39 (2002), 377-385 [9] J Kang, Y Oh, Uniqueness of coexistence state of general competition model for several competing species, Kyungpook Mathematical Journal, 42, No (2002), 391-398 [10] J Kang, Y Oh, J Lee, The existence, nonexistence and uniqueness of global positive coexistence of a nonlinear elliptic biological interacting model, Kangweon-Kyungki Math Jour., 12, No (2004), 77-90 [11] P Korman, A Leung, A general monotone scheme for elliptic systems with applications to ecological models, Proceedings of the Royal Society of Edinburgh, 102A (1986), 315-325 [12] P Korman, A Leung, On the existence and uniqueness of positive steady states in the Volterra-Lotka ecological models with diffusion, Applicable Analysis, 26, 145-160 [13] L Li, R Logan, Positive solutions to general elliptic competition models, Differential and Integral Equations, (1991), 817-834 [14] A Leung, Equilibria and stabilities for competing-species, reactiondiffusion equations with Dirichlet boundary data, J Math Anal Appl., 73 (1980), 204-218 1020 ... inf(hu )φ1 ) with any λ > is a family of subsolutions to (4) POSITIVE EQUILIBRIUM SOLUTIONS TO 1013 Furthermore, if (u, v) is a positive solution to (4), then u > λ0 φ1 and v > λ0 inf(hu )φ1... http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v85i6.4 AP ijpam.eu POSITIVE EQUILIBRIUM SOLUTIONS TO GENERAL POPULATION MODEL Joon Hyuk Kang Department of Mathematics Andrews University Berrien... and g(0, 0) = h(0, 0) = POSITIVE EQUILIBRIUM SOLUTIONS TO 1015 The following theorem provides a sufficient condition for the existence of a positive smooth solution to (6) Theorem 4.2 Suppose