Nonlocal elliptic system arising from the growth of cancer stem cells

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018083 DYNAMICAL SYSTEMS SERIES B Volume 23, Number 4, June 2018 pp. 1767–1795 NONLOCAL ELLIPTIC SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS Manuel Delgado Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Fac. de Matem´aticas, Univ. de Sevilla Sevilla, C/. Tarfia s/n, 41012, Spain ´ Italo Bruno Mendes Duarte Universidade Federal do Par´a Faculdade de Matem´atica Bel´em, PA 66075-110, Brazil Antonio Su´ arez Fern´ andez Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico Fac. de Matem´aticas, Univ. de Sevilla Sevilla, C/. Tarfia s/n, 41012, Spain (Communicated by Thomas Hillen) Abstract. In this work we show the existence of coexistence states for a nonlocal elliptic system arising from the growth of cancer stem cells. For this, we use the bifurcation method and the theory of the fixed point index in cones. Moreover, in some cases we study the behaviour of the coexistence region, depending on the parameters of the problem. 1. Introduction. In this work, we will study the following system    −D1∆u=δγF(u+v)K(u) in Ω, −D2∆v+αv = (1 −δ)γF (u+v)K(u) + ρF (u+v)K(v) in Ω, u=v= 0 on ∂Ω, (1) where Ω is a bounded and regular domain of IRN,D1, D2, γ, α, ρ > 0, δ∈[0,1] and F∈C1(IR+) is a decreasing function with F(0) = 1 and F(t) = 0, for t≥1. The function K(u) : L∞(Ω) −→ L∞(Ω) is given by K(u)(x) = ZΩ K(x, y)u(y)dy, where K∈C(Ω ×Ω) is a non-negative and non-identically zero function. This system is the stationary counterpart, with homogeneous Dirichlet boundary conditions, of a model of the dynamic of cancer stem cells (CSCs) and non-stem tumor cells (TCs) in a certain tissue Ω, proposed in [11]. In that paper, they studied 2010 Mathematics Subject Classification. Primary: 35J60, 35P05; Secondary: 92B05. Key words and phrases. Nonlocal elliptic system, coexistence states, bifurcation method, fixed point index, cancer stem cells. 1767 1768 M. DELGADO, I. B. M. DUARTE AND A. SU´ AREZ FERN´ ANDEZ a particular population of (CSCs). More precisely, the authors studied the following time-dependent system              ∂u(x, t) ∂t =D1∆u+δγ ZΩ K(x, y, p(x, t))u(y, t)dy ∂v(x, t) ∂t =D2∆v−αv +ρZΩ K(x, y, p(x, t))v(y, t)dy +(1 −δ)γZΩ K(x, y, p(x, t))u(y, t)dy, where p(x, t) = u(x, t) + v(x, t), u(x, t) and v(x, t) denote the density, in cells per unit cell space, of cancer stem cells (CSCs) and non-stem cancer cells (TCs) at time tand location x, respectively. The kernel K(x, y, p) describes the rate of progeny contribution to location xfrom a cell at location y, per cell cycle time. The constants D1, D2>0 are diffusion coefficients of the cells (CSCs) and (TCs), respectively. The parameters γ, ρ > 0 denote, respectively, the number of cell cycle times per unit time of (CSCs) and (TCs), and α > 0 denotes the (TCs) death rate. Moreover, δ∈[0,1] denotes the fraction of (CSCs) divisions that are symmetric, that is, the probability in which the cells (CSCs) can give rise to two cells (CSCs), while 1 −δis the fraction of (CSCs) divisions that are not symmetric, that is, the probability in which the cells (CSCs) can give rise to one cell (CSC) and one cell (TC). The boundary conditions of the smooth bounded domain Ω ⊂IRNcan be Dirichlet or Neumann, depending on the tissue Ω. The populations of (CSCs) and (TCs), modeled by the equations above, belong to the class of birth-jump processes, as can be seen in [3]. In a birth-jump process the population growth and spatial spread cannot be decoupled, as is discussed in [15]. These models, of birth-jump processes, are described by the following integral-differential equation ut−d∆u=ZΩ S(x, y, u(x, t))β(u(y, t))u(y, t)dy, (2) where the function Sis the redistribution kernel for the newly generated individuals at yto jump to x, the function β(u) is a proliferation rate at location y. In many situations, S(x, y, u(x, t)) = g(u(x, t))K(x, y),(3) with ga non-negative and non-trivial function, Kis a kernel bounded, non-negative, and depends on xand yjust through of the distance |x−y|. For instance, K(x, y) = ϕ(|x−y|), where ϕ(t) = Ae−Bt2and A,Bare positive constants. Observe that, in this case, we have that K(x, x)>0, for all x∈Ω. We still observe that, in the system (1), g=Fand our choice of Kand Fis motivated by [12]. Now, we observe that there exist three types of solutions of (1): (i) the trivial solution (0,0); (ii) the semi-trivial solutions (u, 0) and (0, v); (iii) the solutions with both positive components, the coexistence states (u, v). The trivial solution always exists. For the existence of semi-trivial solutions, we will introduce some notations and results given in [8]. Observe that when one group of cell vanishes, the other one verifies an equation of the following type:    −d∆u+βu =σF(u)ZΩ K(x, y)u(y)dy in Ω, u= 0 on ∂Ω, (4) NONLOCAL SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS 1769 with β≥0, σ > 0. The problem (4) is a nonlocal logistic equation and has been analyzed in [8] when β= 0 and F(u)=(A(x)−up)+, where p≥1 and A∈C(Ω), with A+6= 0. In Section 2 we study (4) with Fmore general using the sub-super solution method introduced by [8]. Moreover, we study some properties of the solution, as monotonicity in σand uniformly convergence on compacts of Ω when σ→+∞, which will be used throughout this work. For the coexistence states, we will study their existence just in two cases: δ6= 1 and δ= 1. Observe that for δ= 0 the system (1) does not have coexistence states, because in this case (1) is reduced to an equation of the type (4). For the case δ6= 1 we use bifurcation arguments, more precisely [5], [18] and [19], to find an unbounded continuum of coexistence states of (1) emanating from the specific point (see Section 4). Thus, we have the existence of one curve in the plane (γ−ρ), denoted by γ=Fδ(ρ) and we obtain the following result: (a) Assume that δ6= 1, δ6= 0 and ρ > 0. If γ > Fδ(ρ), then there exists at least a coexistence state of (1). Of course we are assuming by biological sense that ρ6= 0, but this result still is true if ρ= 0, as we will see in Section 4 (see Figure 1). For δ= 1, we use the theory present in [1] and [6] of fixed point index with respect to the positive cone and we obtain the existence of two curves γ=F1(ρ) and ρ=G(γ) and show the following results: (b) Assume that δ= 1, γ > σ1,1and ρ > σ1,2(see (9) for more details). Then, there exists a coexistence states if (γ− F1(ρ)) ·(ρ− G(γ)) >0. Depending on relative position of these two curves, we can conclude: (c) Assume that δ= 1, γ > σ1,1and ρ > σ1,2. If γ > F1(ρ) and ρ > G(γ),then there exists at least a coexistence state of (1). Moreover, the sum of the indices of all coexistence states of (1) is 1 (see Figures 2and 3). (d) Assume that δ= 1, γ > σ1,1and ρ>σ1,2. If γ < F1(ρ) and ρ < G(γ),then there exists at least a coexistence state of (1). Moreover, the sum of the indices of all coexistence states of (1) is -1 (see Figures 3and 4). An outline of the paper is: in Section 2 we study the existence of semi-trivial solutions of (1), introducing some notations and results given in [8]. Moreover, we will study the nonlocal logistic problem (4). In Section 3 we study a priori bounds of the coexistence states of (1) and prove results of non existence of coexistence states for (1). In Section 4, we study the existence of coexistence states of (1) in the case that δ6= 1, we use the ideas of [10] applying the Crandall-Rabinowitz Theorem and Theorem 7.2.2 of [19]. In Section 5, we study the existence of coexistence states of (1) when δ= 1, we use the theory of fixed point index with respect to the positive cone introduced by [1] and the ideas present in [20]. Finally, in Section 6, we study the coexistence regions of the solutions of (1) and we analyze in some cases the relative position of the curves γ=F1(ρ) and ρ=G(γ). Firstly, let us fix some notations which we will use along this work. For each u∈Lp(Ω), p∈[1,+∞], |u|pwill denote the usual norm of the space Lp(Ω). For u∈H1 0(Ω), ||u|| will denotes the usual norm of H1 0(Ω). We also consider the following spaces, X=C1 0(Ω) = u∈C1(Ω); u= 0,in ∂Ωand PX={u∈X;u≥0,in Ω}. 1770 M. DELGADO, I. B. M. DUARTE AND A. SU´ AREZ FERN´ ANDEZ Observe that int (PX) = u∈PX;u > 0 in Ω and ∂u ∂η <0 in ∂Ω, where, ηdenotes the outward unit normal vector to ∂Ω. For u∈X,||u||Xwill denote the usual norm of the space X. Moreover, u > 0 means that u∈P\ {0}. 2. Principal eigenvalue and nonlocal logistic equation. In this section, we will study the existence of semi-trivial solutions of (1). Firstly, we will introduce some notations and results of eigenvalue problem given in [8]. Let us see: for d > 0 and m∈L∞(Ω), we will denote by λ1(−d∆ + m(x)) the principal eigenvalue of the problem −d∆u+m(x)u=λu in Ω, u= 0 on ∂Ω, and we will consider the operator Ldefined, in the weak sense, by Lu =−d∆u+m(x)u−ZΩ K(x, y)u(y)dy, ∀u∈X. With that, for the following nonlocal and non self-adjoint eigenvalue problem Lu =λu in Ω, u= 0 on ∂Ω,(5) we have the next result: Proposition 1. Assume that K∈L∞(Ω×Ω) is a non-negative and non-identically zero function, m∈L∞(Ω) and d > 0. Then, there exists a principal eigenvalue of (5), that we will denote by λ1(−d∆ + m(x); K), which is real, simple, it has an associated positive eigenfunction and it is the unique eigenvalue of (5)having an associated eigenfunction without change of sign. Moreover, any other eigenvalue λof (5)satisfies λ1(−d∆ + m(x); K)<Re(λ), the eigenvalue λ1(−d∆ + m(x); K)is the principal eigenvalue of L∗(adjoint of L) and it has the following properties: (i)Let K1, K2∈L∞(Ω ×Ω) be non-negative and non-identically zero functions and m1, m2∈L∞(Ω). If K1≤K2in Ω×Ωand m1≤m2in Ω, then λ1(−d∆ + m1(x); K2)≤λ1(−d∆ + m2(x); K1). Moreover, if K16=K2in Ω×Ωor m16=m2in Ω, the inequality is strict. (ii)Let Ω1,Ω2be regular subdomains of Ω. If Ω1⊂Ω2, then λΩ2 1(−d∆ + m(x); K)≤λΩ1 1(−d∆ + m(x); K). Moreover, if Ω16= Ω2, the inequality is strict. Here, λΩi 1(−d∆ + m(x); K), i= 1,2, denotes the principal eigenvalue of the problem (5)in Ωi. (iii)Let Kn∈L∞(Ω ×Ω) be non-negative and non-identically zero functions. If Kn→Kin L∞(Ω ×Ω), as n→+∞, then λ1(−d∆ + m(x); Kn)→λ1(−d∆ + m(x); K),as n→+∞. Proof. This is proved in Theorem 2.3 and Proposition 2.5 of [8]. NONLOCAL SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS 1771 Now, we will consider the following eigenvalue problem:    −d∆u+m(x)u=σZΩ K(x, y)u(y)dy in Ω, u= 0 on ∂Ω, (6) with σ > 0. Observe that finding an eigenvalue σ1>0 of (6) is equivalent to λ1(−d∆ + m(x); σ1K)=0.(7) We have the next result: Proposition 2. Assume that m∈L∞(Ω) and that λ1(−d∆ + m(x)) >0. Assume further that K∈C(Ω ×Ω) is a non-negative and non-identically zero function. Then, there exists a unique eigenvalue σ1(d;m(x); K)>0of (6). Moreover, the map σ7−→ λ1(−d∆ + m(x); σK)is continuous, decreasing and for all σ > 0,    λ1(−d∆ + m(x); σK)>0,if σ < σ1(d;m(x); K), λ1(−d∆ + m(x); σK)=0,if σ=σ1(d;m(x); K), λ1(−d∆ + m(x); σK)<0,if σ > σ1(d;m(x); K). (8) Proof. See Theorem 2.7 and Proposition 2.6 of [8]. The following corollary will be used to study the behaviour of the coexistence regions of (1). Its proof follows by Propositions 1and 2. Corollary 1. (i)Assume that K1, K2∈C(Ω ×Ω) are non-negative and nonidentically zero functions, m1, m2∈L∞(Ω) and λ1(−d∆ + m1(x)) >0. If K1≤K2in Ω×Ωand m1≤m2in Ω, then σ1(d;m1(x); K2)≤σ1(d;m2(x); K1). Moreover, if K16=K2in Ω×Ωor m16=m2in Ω, the inequality is strict. (ii)Let Kn∈C(Ω ×Ω) be non-negative and non-identically zero functions. If λ1(−d∆ + m(x)) >0and Kn→Kin C(Ω ×Ω), as n→+∞, then σ1(−d;m(x); Kn)→σ1(−d;m(x); K),as n→+∞. For simplicity, in what follows we will make the following notations σ1,1≡σ1(D1; 0; K) and σ1,2≡σ1(D2;α;K).(9) The next characterization will be useful throughout this work, its proof can found in Lemma 2.4 of [8]. Lemma 2.1. The following claims are equivalent: (i)There exists a strict super-solution for (5), that is, there exists a function u∈W2,p(Ω), with p>N, such that u≥0in Ωand satisfies, in the weak sense, −d∆u+m(x)u−ZΩ K(x, y)u(y)dy ≥0in Ω, u ≥0on ∂Ω, with some strict inequality; (ii) (5)verifies the Strong Maximum Principle (SMP); (iii)λ1(−d∆ + m(x); K)>0. In the following proposition, we will prove some properties of λ1(−d∆ + m(x); K) that are not included in [8] and that will be used in this paper. 1772 M. DELGADO, I. B. M. DUARTE AND A. SU´ AREZ FERN´ ANDEZ Proposition 3. (i)If λ1(−d∆ + m(x); K)>0and f∈L2(Ω), then the linear problem:    −∆u+m(x)u−ZΩ K(x, y)u(y)dy =f(x)in Ω, u= 0 on ∂Ω, (10) has unique solution in H1 0(Ω). Moreover, if f∈PX, then u∈PX. Consequently, if λ1(−d∆ + m(x); K)>0and u∈Xsatisfies    −∆u+m(x)u−ZΩ K(x, y)u(y)dy = 0 in Ω, u= 0 on ∂Ω, then u= 0 in Ω. (ii)If λ1(−d∆ + m(x); K)6= 0 and u∈PXsatisfies    −∆u+m(x)u−ZΩ K(x, y)u(y)dy = 0 in Ω, u= 0 on ∂Ω, then u= 0 in Ω. Proof. (i) Consider M > 0 and define the operator T:L2(Ω) −→ L2(Ω) by T(f) = u, where u∈H1 0(Ω) and −∆u+ (m(x) + M)u−ZΩ K(x, y)u(y)dy =f. By Proposition 2.2 of [8], for M > 0 sufficiently large, Tis a well defined, linear and compact operator. Thus, by Fredholm Alternative Theorem, the existence and uniqueness of solution of (10) is equivalent to show that the problem u−MT u = 0 in Ω, u= 0 on ∂Ω,(11) only admits the trivial solution. Since λ1(−d∆ + m(x); K)>0, by Lemma 2.1, (11) only admits the trivial solution. Therefore, (10) has unique solution. Lastly, if f∈PX, the Strong Maximum Principle (see [17]) implies that u∈PX. (ii) By Proposition 1, there exists an eigenfunction ϕ∗ 1>0 of L∗associated to λ1(−d∆ + m(x); K). Hence, 0=(ϕ∗ 1, Lu)=(L∗ϕ∗ 1, u) = λ1(−d∆ + m(x); K)ZΩ ϕ∗ 1u. Since λ1(−d∆ + m(x); K)6= 0 and ϕ∗ 1>0, then u= 0 in Ω. With these notations and results, to search of semi-trivial solutions of (1), we will study the following non-linear problem:    −d∆u+βu =σF(u)ZΩ K(x, y)u(y)dy in Ω, u= 0 on ∂Ω, (12) with β≥0, σ > 0 and Fas in the Introduction. This equation has been analyzed in [8] when β= 0 and F(u)=(A(x)−up)+, where p≥1 and A∈C(Ω), with A+6= 0, but it can be generalized in our case, as we will see in the next proposition. Proposition 4. The following claims about (12)hold: NONLOCAL SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS 1773 (i) (12)has a unique positive solution in X, denoted by θσ[d;β;K], if and only if σ > σ1=σ1(d;β;K). Moreover, θσ[d;β;K]≤1in Ω.(13) (ii)If K1≤K2in Ω×Ωand σ1≤σ2, then θσ1[d;β;K1]≤θσ2[d;β;K2]in Ω. Moreover, the map σ7−→ θσ[d;β;K]is continuous. (iii)Denote θσ[d;β;K]simply by θσ. Then, the principal eigenvalue of the problem −d∆u+βu −σF0(θσ)K(θσ)u−σF(θσ)K(u) = λu in Ω, u= 0 on ∂Ω,(14) is positive, that is, λ1(−d∆ + β−σF 0(θσ(x))K(θσ)(x); σF(θσ(x))K)>0.(15) Proof. (i) Assume first that σ > σ1. We will prove the existence of positive solution of (12) using the sub-super solutions method (Theorem 3.1 of [8]). Let ϕ1>0 be an eigenfunction associated to λ1(−d∆+β;σK). Then, u=ϕ1, with  > 0 sufficiently small, and u= 1 is a pair of sub-super solutions of (12). By Theorem 3.1 of [8], there exists a positive solution u∈Xof (12) such that ϕ1(x)≤u(x)≤1 in Ω. Thus, once proven the uniqueness, (13) follows immediately. Therefore, we will prove the uniqueness of positive solution of (12). For this, suppose that there exist two positive solutions of (12), u6=vin Ω, and let w=u−v. We get,    −d∆w+m(x)w−σF (u)ZΩ K(x, y)w(y)dy = 0 in Ω, w= 0 on ∂Ω, (16) where m(x) = β+σh(x)ZΩ K(x, y)v(y)dy and h(x) =    −F(u)−F(v) u−vif u6=v, −F0(u) if u=v. Hence, (16) implies that there exists j0≥1 such that λj0(−∆ + m(x); σF (u(x))K)=0.(17) On the other hand, observe that vis a strict super-solution for (16). Indeed, it suffices to prove that −d∆v+m(x)v−σF (u)ZΩ K(x, y)v(y)dy > 0 in Ω.(18) Note that −∆v+m(x)v=σF(v)ZΩ K(x, y)v(y)dy +h(x)vZΩ K(x, y)v(y)dy =σ(F(v) + h(x)v)ZΩ K(x, y)v(y)dy. Thus, (18) is equivalent to show σ(F(v) + h(x)v−F(u)) ZΩ K(x, y)v(y)dy>0 in Ω, 1774 M. DELGADO, I. B. M. DUARTE AND A. SU´ AREZ FERN´ ANDEZ that is, we must prove that F(v) + h(x)v−F(u)>0 in Ω.(19) To prove (19), we will see the three possible cases: (a) For the set {x∈Ω; u(x)> v(x)}, we have that u > v ⇒F(v)> F(u) ⇒ −F(u)u+F(v)u > 0 ⇒(F(v)−F(u))(u−v)−F(u)v+F(v)v > 0 ⇒F(v)−F(u)−F(v) u−v·v−F(u)>0 ⇒F(v) + h(x)v−F(u)>0, which proves (19). (b) Similarly to (a), we obtain (19) for the set {x∈Ω; u(x)< v(x)}. (c) For {x∈Ω; u(x) = v(x)}, we have F0(u)<0⇒ −F0(u)>0⇒F(v)−F0(u)v−F(u)>0, which proves (19) in this case. Hence, (18) is true and, from Lemma 2.1, we have that λ1(−∆ + m(x); σF (u(x))K)>0. But this is a contradiction because (17) and Proposition 1imply that 0< λ1(−∆ + m(x); σF (u(x))K)≤Re(λj0(−∆ + m(x); σF(u(x))K)) = 0. Therefore, u=vin Ω. Finally, we show that, if u∈Xis a positive solution of (12), then σ > σ1. Observe that, from Proposition 1, we get λ1(−d∆ + β;σ1K) = 0 = λ1(−d∆ + β;σF (u(x))K)> λ1(−d∆ + β;σK), because uis positive and consequently F(u)<1. From equation (8) in Proposition 2, we have that σ > σ1. (ii) Note that θσ1[d;β;K1] is a sub-solution of (12) for K=K2and σ=σ2. Since u=C, with C > 0 sufficiently large, is a super-solution of (12), (ii) follows by (i). The continuity of the map σ7−→ θσ[d;β;K] follows by (13) and by uniqueness of positive solution for (12). (iii) Since Fis decreasing, then θσis a strict super-solution of the problem (14). Indeed, observe that −d∆θσ+βθσ−σF0(θσ)K(θσ)θσ−σF(θσ)K(θσ) = −σF0(θσ)K(θσ)θσ>0 in Ω. Therefore, (iii) follows by Lemma 2.1. In the next proposition, we show that θσ[d;β;K] converges uniformly to 1, on compacts of Ω, when σ→+∞. For this, we will suppose that, further of the assumptions above, Ksatisfies also that, K(x, x)>0 for all x∈Ω. (20) Proposition 5. Assume (20). The following claim holds: lim σ→+∞θσ[d;β;K]=1,uniformly on compacts of Ω.(21) NONLOCAL SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS 1775 Proof. In this proposition, we will follow the ideas presented in [9], see also [13]. Again we will denote θσ[d;β;K] simply by θσ. To prove (21) we must show that for each compact subset of A⊂Ω and  > 0, there exists σ=σ(A, )>0 such that σ > σ(A, )⇒1−<θσ<1 + in A. First, observe that by (13), θσ≤1 in Ω. Thus, it suffices to prove that σ > σ(A, )⇒θσ>1−in A. (22) Since Ais compact, to show (22) it suffices to show that, given x0∈A, there exists a neighborhood of x0,U0⊂Ω, and a σ1=σ1(x0)>0 such that σ > σ1⇒θσ>1−in U0. Let R > 0 such that B0=BR(x0)⊂Ω. By Proposition 4(i), for σ > 0 sufficiently large, the problem    −d∆u+βu =σF(u)ZΩ∩B0 K(x, y)u(y)dy in B0, u= 0 on ∂B0, (23) has unique positive solution in X, because K(x0, x0)>0. This solution will be denoted by θB0 σ. Since θσis a strict super-solution of (23), then θB0 σ≤θσin B0. Thus, it suffices to show that there exists σ1=σ1(x0)>0 such that σ > σ1⇒θB0 σ>1−in BR1(x0), with R1≤R. Let ϕB0 1>0 be eigenfunction associated to λB0 1(−d∆ + β) such that |ϕB0 1|∞= 1 and ϕB0 1(x0) = 1. Since K(x0, x0)>0, if δ∈(0,1), σ > 0 is sufficiently large and u=δϕB0 1, then we have that λB0 1(−d∆ + β)ϕB0 1≤σF (δϕB0 1)ZΩ K(x, y)ϕB0 1(y)dy in B0, that is, uis a sub-solution of (23). Therefore, since ϕB0 1(x0) = 1 and u= 1 is a super-solution of (23), given  > 0 there exist σ1(x0)>0 and R1≤Rsuch that, for σ > σ1 θB0 σ≥u > 1−in BR1(x0), which finishes the proof. Now, we can study the existence of semi-trivial solution of (1). This study will be divided in two cases: δ6= 1 and δ= 1. For the case δ6= 1, we have the following result: Proposition 6. Assume that δ6= 1. Then: (i) (1)does not have semi-trivial solution of the form (u, 0), with u > 0in Ω. (ii) (1)has semi-trivial solution of the form (0, θρ)if and only if ρ>σ1,2, where θρ≡θρ[D2;α;K].(24) Proof. (i) Suppose that u > 0 in Ω. Then, v= 0 in Ω ⇒0 = (1 −δ)γF (u)ZΩ K(x, y)u(y)dy ⇒F(u) = 0 ⇒u≥1 in Ω, 1782 M. DELGADO, I. B. M. DUARTE AND A. SU´ AREZ FERN´ ANDEZ Thus, dim Ker[D(u,v)F(γ0,0,0)]= 1, because ρ<σ1,2implies that λ1(−D2∆ + α;ρK)>0. Hence, the linear problem −D2∆η+αη = (1 −δ)γK(ϕ1) in Ω, η= 0 on ∂Ω, has unique solution, where ϕ1is an eigenfunction associated to γ0. Moreover, since ρ<σ1,2Proposition 2implies that (ρ, θρ) is a nondegenerate positive solution of (6). Thus, similarly to case ρ>σ1,2, we can use Theorem 7.2.2 of [19] and conclude that there exists a unbounded continuum C+⊂IR ×E1×E2of coexistence states of (1) emanating from the point (γ0,0, θρ) such that (γ0,+∞)⊂Proj(C+). Case ρ=σ1,2:We will study this case by approximation. For this, consider the pair (γ, σ1,2), with δγ > σ1,1. By previous cases, there exists a sequence ρn> σ1,2 such that ρn→σ1,2and there exists solutions (un, vn)∈X×Xof (1) for the parameter γand ρnsuch that un→uand vn→vin X. We must show that u, v > 0. Let ξn=un |un|∞ and ηn=vn |vn|∞ . Observe that |ξn|∞=|ηn|∞= 1. Thus, there exist ξ, η ∈PXsuch that ξn→ξand ηn→η, in X. Suppose that u= 0. Then, −D1∆ξ=δγK(ξ). Since δγ > σ1,1, we have that ξ= 0, a contradiction, because |ξn|∞= 1. Similar to item (ii) above, we can not have v= 0, which concludes the proof of the Theorem. Remark 2. Although we are assuming that ρ > 0 by biological meaning, the above theorem is true if ρ= 0. 5. Coexistence states for the case δ= 1.In this section, we will study the existence of coexistence states of (1) for δ= 1. In this case the system (1) simply is    −D1∆u=γF (u+v)K(u) in Ω, −D2∆v+αv =ρF(u+v)K(v) in Ω, u=v= 0 on ∂Ω. (39) Observe that by Proposition 7, for each γ > σ1,1and ρ>σ1,2, system (39) has the semi-trivial solutions y1= (θγ,0) and y2= (0, θρ). For this case, the a priori bounds given in Proposition 9do not allow us to use bifurcation results again. Thus, we will compute the index of these semi-trivial solutions and of the trivial solution using the theory of fixed point index with respect to the positive cone (see [1], [6] and [20] for more details). For this, we need some notations and results. First, let us consider the sets: N1={u∈PX;u≤ |θγ|∞+ 1 in Ω}, N2={u∈PX;u≤ |θρ|∞+ 1 in Ω} and N=N1×N2. NONLOCAL SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS 1783 Let M > 0 be sufficiently large and define the homotopy H: [0,1] ×N−→ X×X by H(t, u, v)=(L1[Mu +γF(u+tv)K(u)], L2[Mv +ρF(tu +v)K(v)]) , where L1= (−D1∆+M)−1and L2= (−D2∆+α+M)−1, both under homogeneous Dirichlet boundary conditions. Observe that, by choice of N1and N2the homotopy His well defined and admissible. Indeed, if there exists (u0, v0)∈∂N such that H(t, u0, v0)=(u0, v0), for some t∈[0,1], then −∆u0=γF (u0+tv0)K(u0)≤γF(u0)K(u0), and, as in Proposition 9, we get u0≤θγin Ω, which is a contradiction, because (u0, v0)∈∂N. Define E=X×Xand W=PX×PX. We will consider also the following sets: Wy={x∈E;y+tx ∈W, for some t > 0}and Sy=x∈Wy;−x∈Wy. For y1= (θγ,0) and y2= (0, θρ), we have Wy1=X×PX, Wy2=PX×Xand Sy1=X× {0}, Sy2={0} × X. Lastly, let My1={0} × X,My2=X× {0}and consider the continuous projections Py1:E−→ My1and Py2:E−→ My2, given by Py1(u, v) = (0, v) and Py2(u, v)=(u, 0). To compute the total index over Nand the index of trivial solution (0,0), we will use the following lemma. In this lemma, we will consider Pρ=ρB ∩PX, where B is the open unit ball of X, and its proof can be found in [1]: Lemma 5.1. Let f:Pρ−→ PXbe a compact map such that f(0) = 0. Suppose that fhas a right derivative f0 +(0) at zero such that 1 is not an eigenvalue of f0 +(0) to a positive eigenvector. Then, there exists a constant σ0∈(0, ρ]such that for every σ∈(0, σ0], (i)iW(f, Pσ) = 1 if f0 +(0) has no positive eigenvector for an eigenvalue greater than one; (ii)iW(f, Pσ) = 0 if f0 +(0) possesses a positive eigenvector for an eigenvalue greater than one. Remark 3. iW(f, Pρ) denotes the index of fover Pρwith respect to W. More generality, we will denote by iW(T, Z) the index of the operator Tover Zwith respect to the set Z. Moreover, for an isolated fixed point yof the operator T, iW(T, y) will denote the local index of Tat y(see [1] for more details). The next lemma will be used to compute the index of semi-trivial solutions. Its proof can be found in [6]: Lemma 5.2. (i)If I−DxH(1, y)is an invertible operator on Eand the spectral radius of PyDxH(1, y)|My, denoted by Spr (PyDxH(1, y)|My), is greater than one, then iW(H(1,·), y)=0. (ii)If I−DxH(1, y)is an invertible operator on Eand Spr (PyDxH(1, y)|My)<1, then iW(H(1,·), y)=(−1)χ, where χis the sum of the multiplicities of the all eigenvalues of DxH(1, y)greater than one. (iii)If I−DxH(1, y)is an invertible on Wyinstead of Eand there is some w∈Wy such that the equation (I−DxH(1, y))x=whas no solution x∈Wy, then iW(H(1,·), y)=0. 1784 M. DELGADO, I. B. M. DUARTE AND A. SU´ AREZ FERN´ ANDEZ We use several times the following lemma: Lemma 5.3. Assume that Tis a compact and strongly positive linear operator on an ordered Banach space e X, with int (Pe X)6=∅. Let u > 0be a positive element of e X. We have the following conclusions: (i)If Tu > u, then Spr T > 1. (ii)If Tu < u, then Spr T < 1. (iii)If Tu =u, then Spr T= 1. Proof. We will prove the item (i). Assume that u−Tu < 0. Since Tis a strongly positive linear operator, then Tis a positive irreducible operator. Moreover, Tis a compact and int (Pe X)6=∅. It follows by the Theorem 12.3 of [7] that r(T) = Spr T is a simple eigenvalue of T∗with a strictly positive eigenfunction associated. Then, 0>(u, u∗)−(Tu, u∗)=(u, u∗)−(u, Tu∗)=(u, u∗)−r(T)(u, u∗), where we deduce that Spr T=r(T)>1, because (u, u∗)>0. The items (ii) and (iii) follow analogously. Remark 4. A similar result has been proved in [16] assuming that Pe Xis a normal cone and int (Pe X)6=∅because the classical Krein-Rutman Theorem is used (see [1]). However, we are going to use the result for the space C1 0(Ω) where the cone is not normal (see [1]). With these considerations, we have the following result: Theorem 5.4. Assume that γ > σ1,1and ρ > σ1,2, then the following claims are verified: (i)iW(H(1,·,·), N) = 1; (ii) (0,0) is an isolated solution of H(1,·,·), moreover, iW(H(1,·,·),(0,0)) = 0; (iii)iW(H(1,·,·),(0, θρ)) = 0, if γ > σ1(D1; 0; F(θρ(x))K); (iv)iW(H(1,·,·),(0, θρ)) = 1, if γ < σ1(D1; 0; F(θρ(x))K); (v)iW(H(1,·,·),(θγ,0)) = 0, if ρ>σ1(D2;α;F(θγ(x))K); (vi)iW(H(1,·,·),(θγ,0)) = 1, if ρ<σ1(D2;α;F(θγ(x))K). Proof. (i) From the properties of index, iW(H(1,·,·), N) = iW(H(0,·,·), N) = 2 Y j=1 iPX(Hj, Nj), where iPX(Hj, Nj) is the index of Hjover Njwith respect to PXand H1(u) = L1[Mu +γF(u)K(u)] and H2(v) = L2[Mv +ρF(v)K(v)]. We will show that iPX(H1, N1) = iPX(H2, N2)=1. For this, we fix M > 0 and we define the homotopies G1(t, u) = L1(Mu +tγF(u)K(u)) , G2(t, v) = L2(Mv +tρF(v)K(v)) , for each (t, u, v)∈[0,1] ×N. From the homotopy invariance property, we get iPX(Hj, Nj) = iPX(Gj(1,·), Nj) = iPX(Gj(0,·), Nj) = iPX(Gj(0,·),0), NONLOCAL SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS 1785 for j= 1,2. Now, observe that Spr G1(0,·)<1 and Spr G2(0,·)<1. Indeed, for instance, if r∈IR is such that G1(0, u) = ru, with u∈N1and u6= 0, then −D1∆u=M1 r−1u. Since λ1(−D1∆) >0, then r < 1. Similarly for G2(0,·), because α > 0. Hence, (i) follows by Lemma 5.1. (ii) Observe that D(u,v)H(1,0,0)(u, v) = L1(Mu +γK(u)) L2(Mv +ρK(v)) . Since γ > σ1,1and ρ>σ1,2, the operator I−D(u,v)H(1,0,0) is invertible on W, that is, 1 is not eigenvalue of D(u,v)H(1,0,0) with a positive eigenfunction. We will show that the operator T:PX→PXdefined by Tu =L1(Mu+γK(u)) has spectral radius greater than one. For this, observe that since M > 0 is sufficiently large, we can use the arguments in Proposition 3(i) and the Strong Maximum Principle of [17] and conclude that the operator Tis a compact and strongly positive linear operator. On the other hand, since γ > σ1,1, by Proposition 2, there exists µ∈(σ1,1, γ) such that λ1(−D1∆; µK)<0. Let ϕ1>0 be an eigenfunction associated to λ1(−D1∆; µK), then Tϕ1> ϕ1. Indeed, we have that Tϕ1> ϕ1⇔L1(Mϕ1+γK(ϕ1)) > ϕ1 ⇔γK(ϕ1)> λ1(−D1∆; µK)ϕ1+µK(ϕ1) ⇔(γ−µ)K(ϕ1)> λ1(−D1∆; µK)ϕ1. Since λ1(−D1∆; µK)<0 and γ > µ, then Tϕ1> ϕ1. By Lemma 5.3, we have that r1= Spr T > 1. Let Ψ1>0 be an eigenfunction associated to r1. Then, D(u,v)H(1,0,0)(Ψ1,0) = r1(Ψ1,0), where (ii) follows by Lemma 5.1. (v) Observe that once proven (v), (iii) follows by symmetry. Thus, we will prove only (v). We have that D(u,v)H(1, θγ,0)(u, v) = L1[Mu +γF0(θγ)K(θγ)(u+v) + γF(θγ)K(u)] L2[Mv +ρF (θγ)K(v)] . By the Maximum Principle of [17], the operator D(u,v)H(1, θγ,0) maps Wy1into Wy1. We will show that I−D(u,v)H(1, θγ,0) is invertible on Wy1. For this, let (u, v)∈Wy1such that (I−D(u,v)H(1, θγ,0))(u, v) = (0,0). Since ρ>σ1(D2;α;F(θγ(x))K), Proposition 3(iii) implies that v= 0. Hence, −D1∆u−γF 0(θγ)K(θγ)u−γF(θγ)K(u)=0. On the other hand, from Proposition 4(iii) λ1(−D1∆−γF 0(θγ(x))K(θγ)(x); γF(θγ(x))K)>0. Thus, by Proposition 3(i) we have that u= 0. Hence, I−D(u,v)H(1, θγ,0) is invertible on Wy1. Now, from the Lemma 5.2it suffices to prove that the operator I−D(u,v)H(1, θγ,0) does not have full rank. Suppose that the operator has full 1786 M. DELGADO, I. B. M. DUARTE AND A. SU´ AREZ FERN´ ANDEZ rank. Then, take v0∈PX\ {0}such that L2v0∈PX, then there exists v∈PXand f∈Xsatisfying (I−D(u,v)H(1, θγ,0))(u, v)=(f, L2v0), that is, −D2∆v+αv −ρF(θγ)ZΩ K(x, y)v(y)dy =v0>0. Thus, by Lemma 2.1, we have ρ<σ1(D2;α;F(θγ(x))K), a contradiction. Therefore, I−D(u,v)H(1, θγ,0) does not have full rank, and (v) follows by Lemma 5.2. (vi) Observe that once proven (vi), (iv) follows by symmetry. Thus, we will prove only (vi). We have that the operator I−D(u,v)H(1, θγ,0) is invertible on E. Indeed, let (u, v)∈Esuch that (I−D(u,v)H(1, θγ,0))(u, v) = (0,0). Once again by Proposition 3(iii) we have that v= 0 and −D1∆u−γF 0(θγ)K(θγ)u−γF(θγ)K(u) = 0 ⇒u= 0. Now, we will show that Spr Py1D(u,v)H(1, θγ,0)|My1<1. This is equivalent to show that the operator T:X→Xdefined by Tv =L2(Mv +ρF(θγ)K(v)) has spectral radius less than 1. For this, observe that again Tis a compact and strongly positive linear operator. On the other hand, since ρ<σ1(D2;α;F(θγ(x))K), there exists µ∈(ρ, σ1(D2;α;F(θγ(x))K)) such that λ1(−D2∆+α;µF(θγ(x))K)>0. If ϕ1>0 is an eigenfunction associated λ1(−D2∆ + α;µF(θγ(x))K), then Tϕ1< ϕ1. Indeed, we have that Tϕ1< ϕ1⇔L2(Mϕ1+ρF(θγ)K(ϕ1)) < ϕ1 ⇔Mϕ1+ρF (θγ)K(ϕ1)<−D2∆ϕ1+αϕ1+Mϕ1 ⇔(ρ−µ)F(θγ)K(ϕ1)< λ1(−D2∆ + α;µF(θγ(x))K)ϕ1. Since λ1(−D2∆ + α;µF(θγ(x))K)>0 and ρ < µ, then Tϕ1< ϕ1. Hence, by Lemma 5.3, we have that Spr Py1D(u,v)H(1, θγ,0)|My1<1. Lastly, we will show that χ= 0, where χis the sum of the multiplicities of the all eigenvalues of D(u,v)H(1, θγ,0) greater than one. Let λbe an eigenvalue of D(u,v)H(1, θγ,0) with eigenfunction (u0, v0). We have two alternatives, v06= 0 or v0= 0. For v06= 0, λis an eigenvalue of Py1D(u,v)H(1, θγ,0)|My1. Since Py1D(u,v)H(1, θγ,0)|My1has spectral radius less than 1, we get λ < 1 and χ=0, in this case. For v0= 0, we have u06= 0. Hence, L1(Mu0+γF0(θγ)K(θγ)u0+γF(θγ)K(u0)) = λu0. NONLOCAL SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS 1787 It suffices to show that the spectral radius of the operator T:X−→ X, defined by T(u) = L1(Mu +γF0(θγ)K(θγ)u+γF(θγ)K(u)) , is less than 1. For this, consider m: Ω −→ IR defined by m(x) = −γF 0(θγ)ZΩ K(x, y)θγ(y)dy. Observe that the operator Tis strongly positive. Indeed, let f∈PXand u=T(f), we get (−D1∆ + M)u= (M−m(x))f(x) + γF (θγ)ZΩ K(x, y)f(y)dy. Since M > 0 is sufficiently large, the Maximum Principle of [17] gives us that T(f) = u∈int PX, that is, Tis strongly positive. Moreover, Tis a compact and linear operator. On the other hand, if ϕ1>0 is an eigenfunction associated to λ1(−D1∆ + m(x); γF (θγ(x))K). Observe that (−D1∆ + M)ϕ1=λ1(−D1∆ + m(x); γF (θγ)(x)K)ϕ1+ (M−m(x))ϕ1 +γF (θγ)ZΩ K(x, y)ϕ1(y)dy > Mϕ1−m(x)ϕ1+γF(θγ)ZΩ K(x, y)ϕ1(y)dy, that is, ϕ1> T (ϕ1). By Lemma 5.3, we have that Spr T < 1, hence, λ < 1 and χ=0, also in this case. Therefore, (v) follows by Lemma 5.2. As consequence of Theorem 5.4, we have the following results: Theorem 5.5. Assume that δ= 1,γ > σ1,1and ρ>σ1,2. If (γ−σ1(D1; 0; F(θρ(x))K)) ·(ρ−σ1(D2;α;F(θγ(x))K)) >0,(40) then there exists at least a coexistence state of (39). Corollary 2. Assume that δ= 1,γ > σ1,1and ρ>σ1,2. If γ > σ1(D1; 0; F(θρ(x))K)and ρ>σ1(D2;α;F(θγ(x))K), then there exists at least a coexistence state of (39). Moreover, the sum of the index of all coexistence states of (39)is 1. Corollary 3. Assume that δ= 1,γ > σ1,1and ρ>σ1,2. If γ < σ1(D1; 0; F(θρ(x))K)and ρ<σ1(D2;α;F(θγ(x))K), then there exists at least a coexistence state of (39). Moreover, the sum of the index of all coexistence states of (39)is -1. Remark 5. 1. Recall that when an isolated solution has index 1 (resp. -1), it is generically stable (resp. unstable) with respect to its associated parabolic problem, see for instance [14]. 2. Observe that, in all the results of Sections 5 and 6, the compactness of some operators has been essential. Hence, we need the coefficients D1and D2are both positive. When D1and/or D2vanish, the integral term is in fact a nonlocal diffusion term (see Remark 2.2 in [12]) and it is a very interesting problem to study the stationary solutions in such case. 1788 M. DELGADO, I. B. M. DUARTE AND A. SU´ AREZ FERN´ ANDEZ 6. Coexistence regions. In this section, we analyze the coexistence regions of (1), that is, the region of the plane (γ−ρ)⊂IR2defined by (32) when δ6= 1 and by (40) when δ= 1. For this, we need to suppose again that Ksatisfies (20), that is, K(x, x)>0 for all x∈Ω. Firstly, let us see the following result of convergence: Proposition 11. The following limit holds lim σ→+∞σ1(d;β;F(θσ[d;β;K](x))K) = +∞. Proof. We will denote θσ[d;β;K] simply by θσ. Suppose that there exists M > 0 such that σn=σ1(−d∆ + β;F(θn(x))K)≤M, for all n>σ1(d;β;K). Since F≤1, we have 0 = λ1(−d∆ + β;σnF(θn(x))K) ≥λ1(−d∆ + β;MF (θn(x))K) ≥λ1(−d∆ + β;MK). Thus, the sequence {λ1(−d∆ + β;MF(θn(x))K)}n∈IN is bounded. Let ϕn>0 be eigenfunction associated to λ1(−d∆ + β;MF(θn(x))K), with |ϕn|2= 1. Then, for each ϕ∈C∞ 0(Ω), we get dZΩ ∇ϕn· ∇ϕ+βZΩ ϕnϕ=λ1(−d∆ + β;MF (θn(x))K)ZΩ ϕnϕ −ZΩ MF (θn(x)) ZΩ K(x, y)ϕn(y)dyϕ(x)dx. Taking ϕ=ϕnwe have that {ϕn}is bounded in H1 0(Ω). Thus, making λn=λ1(−d∆ + β;MF (θn(x))K), up to a subsequence if necessary,    λn→λ∗ 1,in IR ϕn* ϕ∗,in H1 0(Ω) ϕn→ϕ∗,in L2(Ω), with ϕ∗ 1≥0 in Ω and |ϕ∗ 1|2= 1. By Proposition 5,θn→1 uniformly on each compact subset of A⊂Ω, as n→+∞. Thus, F(θn)→0 uniformly on each compact subset of A⊂Ω, as n→+∞. Hence, for all ϕ∈C∞ 0(Ω), we have ZΩ MF (θn(x)) ZΩ K(x, y)ϕn(y)dyϕ(x)dx →0,as n→+∞, which implies, taking limit in the last equation, that dZΩ ∇ϕ∗· ∇ϕ+βZΩ ϕ∗ϕ=λ∗ 1ZΩ ϕ∗ϕ, ∀ϕ∈C∞ 0(Ω). Since C∞ 0(Ω) is dense in H1 0(Ω), we have dZΩ ∇ϕ∗· ∇ϕ+βZΩ ϕ∗ϕ=λ∗ 1ZΩ ϕ∗ϕ, ∀ϕ∈H1 0(Ω). Therefore, λ∗ 1=λ1(−d∆ + β)>0. But, this is a contradiction, because λn→λ∗ 1 and λn=λ1(−d∆ + β;MF (θn(x))K)≤0, NONLOCAL SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS 1789 which concludes the proof. Now, consider δ6= 1 and the function Fδ: [σ1,2,+∞)→[σ1,1 δ,+∞) defined by Fδ(ρ) = σ1(D1; 0; δF(θρ(x))K). The function Fδhas the following properties: Proposition 12. The following claims hold: (i)Fδ(σ1,2) = σ1,1 δ; (ii)The map ρ→ Fδ(ρ)is continuous and increasing; (iii) lim ρ→+∞Fδ(ρ)=+∞; (iv) lim δ→1Fδ(ρ) = F1(ρ), if ρ∈Λ, with Λ⊂IR compact subset; (v)Fδ(ρ)>F1(ρ), for all ρ∈[σ1,2,+∞)and δ < 1. Proof. To prove (i) observe that, for ρ=σ1,2, Proposition 4(i) implies that θρ= 0. Hence, F(θρ) = 1 and Fδ(σ1,2) = σ1(D1; 0; δK) = σ1,1 δ. Item (ii) is proven in Proposition 4(ii). Moreover, (iii) follows immediately from Proposition 11. The items (iv) and (v) follow by Corollary 1. Hence, for the case δ6= 1, we have the coexistence region of (1) given in the Figure 1. ρ σ1,2 σ1,1 δ γ = ℱδ(ρ) γ Figure 1. Coexistence region of (1) for δ6= 1. For δ= 1, we will consider the function G: [σ1,1,+∞)→[σ1,2,+∞) defined by G(γ) = σ1(D2;α;F(θγ(x))K). Similarly to Proposition 12, the function Ghas the following properties: Proposition 13. The following claims hold: (i)G(σ1,1) = σ1,2; 1790 M. DELGADO, I. B. M. DUARTE AND A. SU´ AREZ FERN´ ANDEZ (ii)The map γ→ G(γ)is continuous and increasing; (iii) lim γ→+∞G(γ) = +∞. Remark 6. We can also study the dependence of the above functions Fδ(ρ) and G(γ) with respect to the parameter α. With a similar proof to Proposition 12, we can show: 1. σ1,2is a continuous and increasing function on αand σ1,2→ ∞ as α→ ∞. 2. Fix δ∈[0,1] and ρ>σ1,2, then Fδ(ρ) decreases as αincreases, and Fδ(ρ) = σ1,1/δ for αlarge. 3. Fix γ > σ1,1, then G(γ) increases as αincreases, and G(γ)→+∞as α→+∞. Hence, for δ= 1, we have the following possible coexistence regions of (1) in the Figures 2,3and 4. ρ σ1,2 σ1,1 γ = ℱ1(ρ) γ ρ = 𝒢(γ) Figure 2. Possible coexistence region of (1) for δ= 1. In this case the sum of the index of the coexistence states of (1) is 1. Remark 7. We will analyze the coexistence regions of (1). Denote by Cδand C1 the coexistence regions of (1) (Theorem 4.1and Theorem 5.5) for 0 < δ < 1 and δ= 1, respectively, that is, Cδ=(γ, ρ)∈IR2;ρ > 0 and γ > Fδ(ρ)) and C1=(γ, ρ)∈IR2; (γ− F1(ρ)) ·(ρ− G(γ)) >0. Moreover, denote by Eδand E1the extinction sets of (1) (Proposition 10): Eδ=n(γ, ρ)∈IR2;γ≤σ1,1 δo,for δ6= 1 and E1=(γ, ρ)∈IR2;γ≤σ1,1and ρ≤σ1,2,for δ= 1. In Figure 1we have represented Cδand in Figures 2,3and 4different possibilities of C1. We will first analyze the case 0 < δ < 1. Observe that Eδ→IR2as δ→0 (see Figure 5), hence for any γ > 0 and ρ > 0 there exists δ0such that if δ≤δ0 NONLOCAL SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS 1791 ρ σ1,2 σ1,1 γ = ℱ1(ρ) γ ρ = 𝒢(γ) Figure 3. Possible coexistence region of (1) for δ= 1. In this case the sum of the index of the coexistence states of (1) is -1. ρ σ1,2 σ1,1 γ = ℱ1(ρ) γ ρ = 𝒢(γ) Figure 4. Possible coexistence region of (1) for δ= 1. In this case, there are regions where the sum of the index of the coexistence states of (1) is 1 (when F1is above G) and others where the sum is -1 (when Gis above F1). both species do not coexist. Thus, only the trivial solution (u, v) = (0,0) and the semi-trivial solution (u, v) = (0, θρ) exist for (1) (this last solution if ρ>σ1,2). This is a logical sense: if δis small the cells (CSCs) divide in one cell (CSC) and one cell (TC), and then (CSCs) is driven to the extinction. However, fixed γ > 0, for δγ > σ1,1there exists coexistence states for ρ∈(0, ρ0(δ)), where γ=Fδ(ρ0(δ)) (see Figure 1). For the δ= 1, in this case the cell (CSCs) divide in two cells (CSCs),