NONLOCAL ELLIPTIC SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS

. 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.

(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    −D 1 ∆u = δγF (u + v) K(u) in Ω, −D 2 ∆v + αv = (1 − δ)γF (u + v)K(u) + ρF (u + v) K(v) in Ω, where Ω is a bounded and regular domain of IR N , D 1 , D 2 , γ, α, ρ > 0, δ ∈ [0, 1] and F ∈ C 1 (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 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 a particular population of (CSCs). More precisely, the authors studied the following time-dependent system ∂u(x, t) ∂t = D 1 ∆u + δγ Ω K(x, y, p(x, t))u(y, t)dy ∂v(x, t) ∂t = D 2 ∆v − αv + ρ Ω K(x, y, p(x, t))v(y, t)dy 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 t and location x, respectively. The kernel K(x, y, p) describes the rate of progeny contribution to location x from a cell at location y, per cell cycle time.
The constants D 1 , D 2 > 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 − δ is 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 ⊂ IR N can 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 S(x, y, u(x, t))β(u(y, t))u(y, t)dy, where the function S is the redistribution kernel for the newly generated individuals at y to 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), with g a non-negative and non-trivial function, K is a kernel bounded, non-negative, and depends on x and y just through of the distance |x − y|. For instance, where ϕ(t) = Ae −Bt 2 and A, B are 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 = F and our choice of K and F is 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) Ω K(x, y)u(y)dy in Ω,

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) − u p ) + , where p ≥ 1 and A ∈ C(Ω), with A + = 0. In Section 2 we study (4) with F more general using the sub-super solution method introduced by [8]. Moreover, we study some properties of the solution, as monotonicity in σ and 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: δ = 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).
Of course we are assuming by biological sense that ρ = 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 = F 1 (ρ) and ρ = G(γ) and show the following results: (b) Assume that δ = 1, γ > σ 1,1 and ρ > σ 1,2 (see (9) for more details). Then, there exists a coexistence states if Depending on relative position of these two curves, we can conclude: (c) Assume that δ = 1, γ > σ 1,1 and ρ > σ 1,2 . If γ > F 1 (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 2 and 3).
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 δ = 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 = F 1 (ρ) and ρ = G(γ).
Observe that int (P X ) = u ∈ P X ; u > 0 in Ω and ∂u ∂η < 0 in ∂Ω , where, η denotes the outward unit normal vector to ∂Ω. For u ∈ X, ||u|| X will 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 λ 1 (−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 L defined, in the weak sense, by With that, for the following nonlocal and non self-adjoint eigenvalue problem 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 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 λ of (5) satisfies the eigenvalue λ 1 (−d∆ + m(x); K) is the principal eigenvalue of L * (adjoint of L) and it has the following properties: (i) Let K 1 , K 2 ∈ L ∞ (Ω × Ω) be non-negative and non-identically zero functions and m 1 , m 2 ∈ L ∞ (Ω). If K 1 ≤ K 2 in Ω × Ω and m 1 ≤ m 2 in Ω, then . Moreover, if Ω 1 = Ω 2 , the inequality is strict. Here, λ Ωi 1 (−d∆ + m(x); K), i = 1, 2, denotes the principal eigenvalue of the problem (5) in Ω i . (iii) Let K n ∈ L ∞ (Ω × Ω) be non-negative and non-identically zero functions. If K n → K in L ∞ (Ω × Ω), as n → +∞, then Proof. This is proved in Theorem 2.3 and Proposition 2.5 of [8].
The following corollary will be used to study the behaviour of the coexistence regions of (1). Its proof follows by Propositions 1 and 2.
The next characterization will be useful throughout this work, its proof can found in Lemma 2.4 of [8].
With these notations and results, to search of semi-trivial solutions of (1), we will study the following non-linear problem: with β ≥ 0, σ > 0 and F as in the Introduction. This equation has been analyzed in [8] when β = 0 and F (u) = (A(x) − u p ) + , where p ≥ 1 and A ∈ C(Ω), with A + = 0, but it can be generalized in our case, as we will see in the next proposition.
simply by θ σ . Then, the principal eigenvalue of the problem is positive, that is, Proof. (i) Assume first that σ > σ 1 . We will prove the existence of positive solution of (12) using the sub-super solutions method (Theorem 3.1 of [8]). Let ϕ 1 > 0 be an eigenfunction associated to λ 1 (−d∆+β; σK). Then, u = ϕ 1 , 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 ∈ X of (12) such that 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), u = v in Ω, and let w = u − v. We get, where Hence, (16) implies that there exists j 0 ≥ 1 such that On the other hand, observe that v is a strict super-solution for (16). Indeed, it suffices to prove that Note that Thus, (18) is equivalent to show that is, we must prove that To prove (19), we will see the three possible cases: which proves (19) in this case.
Hence, (18) is true and, from Lemma 2.1, we have that But this is a contradiction because (17) and Proposition 1 imply that Therefore, u = v in Ω. Finally, we show that, if u ∈ X is a positive solution of (12), then σ > σ 1 . Observe that, from Proposition 1, we get because u is positive and consequently F (u) < 1. From equation (8) in Proposition 2, we have that σ > σ 1 .
In the next proposition, we show that θ σ [d; β; K] converges uniformly to 1, on compacts of Ω, when σ → +∞. For this, we will suppose that, further of the assumptions above, K satisfies also that, Proposition 5. Assume (20). The following claim holds: Proof. In this proposition, we will follow the ideas presented in [9], see also [13]. Again we will denote θ σ [d; β; K] simply by θ σ . To prove (21) we must show that for each compact subset of A ⊂ Ω and > 0, there exists σ = σ(A, ) > 0 such that First, observe that by (13), θ σ ≤ 1 in Ω. Thus, it suffices to prove that Since A is compact, to show (22) it suffices to show that, given has unique positive solution in X, because K(x 0 , x 0 ) > 0. This solution will be denoted by θ B0 σ . Since θ σ is a strict super-solution of (23), then that is, u is a sub-solution of (23). Therefore, since ϕ B0 1 (x 0 ) = 1 and u = 1 is a super-solution of (23), given > 0 there exist σ 1 (x 0 ) > 0 and R 1 ≤ R such that, , which finishes the proof. Now, we can study the existence of semi-trivial solution of (1). This study will be divided in two cases: δ = 1 and δ = 1. For the case δ = 1, we have the following result: Proposition 6. Assume that δ = 1. Then: Proof. (i) Suppose that u > 0 in Ω. Then, that is, u = 0 on ∂Ω. Thus, (u, 0) is not semi-trivial solution of (1).
Finally, we conclude this section by studying the following perturbation of the problem (12) which will be used in the next section: with B ∈ C(Ω) a non-negative and non-identically zero function. We have the next proposition: is a non-negative and non-identically zero function. Then, (27) has a unique positive solution, which will be denoted by Proof. The existence follows similarly to item (i) of Proposition 4, with u = 0 and u = Ce, where e > 0 is the unique solution of the problem Ω ⊂ IR N is a regular domain, with Ω ⊂ Ω, and C > 0 is sufficiently large such that

NONLOCAL SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS 1777
For the uniqueness, suppose that there exist two positive solutions of (27), u = v in Ω, and let w = u − v. Hence, that is, the uniqueness also follows similarly to item (i) of Proposition 4.

3.
A priori bounds and non-existence results. In this section, we will study a priori bounds of the coexistence states of (1) and prove non-existence results of coexistence states for (1). Let us start with a priori bounds. and where Proof. For (28) it suffices to note that Since u = 1 is a super-solution of (30), then (28) follows by Proposition 4(ii). On the other hand, by (28) we have Hence, (29) follows similarly to (28), using Proposition 8 and equation (25).
We have the following results about non-existence of coexistence states of (1), which follows immediately of Proposition 4(i). (ii) If δ = 1 and ρ ≤ σ 1,2 , then (1) does not have coexistence states.

4.
Coexistence states for the case δ = 1. In this section, we will study the existence of coexistence states of (1) in the case that δ = 1. Observe that if δ = 0 then (1) implies that u = 0. Hence, in this case, (1) does not have coexistence states. Thus, in this section, we assume that δ = 0.
We are going to apply the bifurcation method in this section. Let us point out some important remarks in the application of the bifurcation results to elliptic systems: 1. In order to apply the classical Rabinowitz's Theorem [22] we need to write our system (1) in the form where U = (u, v) ∈ E := E 1 × E 2 , E i Banach spaces, K is a compact linear operator in E, N (λ, U ) a continuos operator, compact on bounded sets, such that N (λ, U ) = o( U ) as U → 0 uniformly in any compact interval of IR and λ ∈ IR the bifurcation parameter. However, our system can not be written in this way, because we have different parameters in our equations. 2. Observe that if we could apply the Rabinowitz's Theorem, the continuum of nontrivial solutions emanating from the trivial solution could be a semi-trivial solution (u, 0) or (0, v), i. e., it might not contain coexistence states. 3. To overcome these difficulties Blat and Brown [2] act of the following way: Fix the parameter ρ, bifurcate from the semi-trivial solution (0, θ ρ ) and consider γ as bifurcation parameter. Following this strategy, in [19] an abstract theory is developed to show the existence of a continuum of coexistence states emanating from a semi-trivial solution. 4. First, we localize a value of γ, γ 0 , such that the fixed point index of (0, θ ρ ) changes sign as γ crosses γ 0 . Mainly, we apply the Crandall-Rabinowitz Theorem to find the value of γ 0 = σ 1 (D 1 ; 0; δF (θ ρ (x))K). 5. As consequence of this change of index, there exists a continuum Σ of nontrivial solutions, which possesses a subcontinuum Σ + such that in a neighborhood of (γ 0 , 0, θ ρ ) are coexistence states. Let C + denote the subcomponent of where P i is the positive cone of E i . 6. This continuum has two possibilities: or it is unbounded in IR × E 1 × E 2 or it leaves int(P 1 ) × int(P 2 ). If the second option occurs, then: (a) or it leaves int(P 1 ) × int(P 2 ) across ∂P 1 , in such case there exists γ 1 such that (γ 1 , 0, v γ1 ) ∈ cl(C + ), where cl(C + ) denotes the closure of the set C + ; (b) or it leaves int(P 1 ) × int(P 2 ) across ∂P 2 , in such case there exists γ 2 such that (γ 2 , u γ2 , 0) ∈ cl(C + ); (c) or there exists γ 3 such that (γ 3 , 0, 0) ∈ cl(C + ). 7. We have to decide which possibility occurs. Remark 1. By Proposition 4, for 0 < ρ ≤ σ 1,2 , we have θ ρ ≡ 0. Thus, We will use this in the next result.

NONLOCAL SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS 1779
then there exists at least a coexistence state of (1).
Proof. We will apply the Crandall-Rabinowitz Theorem (see [5]) considering γ as bifurcation parameter and we will prove the existence of a value of γ, γ 0 , which determines a bifurcation point from the semi-trivial solution (0, θ ρ ) for each ρ > σ 1,2 and from the trivial solution (0, 0) for 0 < ρ < σ 1,2 , the case ρ = σ 1,2 will result by approximation. First, we will introduce some notation given in [19]. Denote by e 1 and e 2 , respectively, the unique positive solutions of the following linear problems: and −D 2 ∆e 2 + αe 2 = 1 in Ω, Observe that e i ∈ X are strictly positive functions, for i = 1, 2. Let E i , i = 1, 2, denote the Banach space consisting of all functions w ∈ C(Ω) for which there exists endowed with the norm Then E i is an ordered Banach space whose positive cone, denoted by P i , is normal and has a nonempty interior. Moreover, E i → C(Ω) (see [1] and [19] for more details). Now, we will study each case said above. Let us see: Case ρ > σ 1,2 : Consider the operator where L 1 = (−D 1 ∆) −1 and L 2 = (−D 2 ∆ + α) −1 under homogeneous Dirichlet boundary conditions. We have that the operator F is well defined and We claim that, for γ 0 = σ 1 (D 1 ; 0; δF (θ ρ (x))K), To prove this, let us consider ϕ 1 eigenfunction associated to γ 0 . Observe that by Proposition 3(i) and Proposition 4(iii), the linear problem has a unique solution, because This solution will be denoted by ϕ 2 . Observe also that Therefore, by (37) and (36), we have On the other hand, differentiating with respect to γ, we obtain .
We must show that For this, suppose that there exists (ξ, η) ∈ X × X such that Let L * be the adjoint of the operator L : X → X defined by K(x, y)u(y)dy.

NONLOCAL SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS 1781
We will show that each of the last three items can not occur: (ii) Suppose that there exists (γ n , u n , v n ) ∈ C + such that (γ n , u n , v n ) → (γ 1 , u γ 1 , 0) in C + .
As above, there exist ξ, η ∈ P X such that ξ n → ξ and η n → η, in X.

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 Observe that by Proposition 7, for each γ > σ 1,1 and ρ > σ 1,2 , system (39) has the semi-trivial solutions y 1 = (θ γ , 0) and y 2 = (0, θ ρ ). For this case, the a priori bounds given in Proposition 9 do 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:

NONLOCAL SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS 1783
Let M > 0 be sufficiently large and define the homotopy H : and, as in Proposition 9, we get u 0 ≤ θ γ in Ω, which is a contradiction, because (u 0 , v 0 ) ∈ ∂N . Define E = X × X and W = P X × P X .
We will consider also the following sets: For y 1 = (θ γ , 0) and y 2 = (0, θ ρ ), we have Lastly, let M y1 = {0} × X, M y2 = X × {0} and consider the continuous projections P y1 : E −→ M y1 and P y2 : E −→ M y2 , given by To compute the total index over N and the index of trivial solution (0, 0), we will use the following lemma. In this lemma, we will consider P ρ = ρB ∩ P X , where B is the open unit ball of X, and its proof can be found in [1]: Lemma 5.1. Let f : P ρ −→ P X be a compact map such that f (0) = 0. Suppose that f has a right derivative f + (0) at zero such that 1 is not an eigenvalue of f + (0) to a positive eigenvector. Then, there exists a constant σ 0 ∈ (0, ρ] such that for every σ ∈ (0, σ 0 ], (i) i W (f, P σ ) = 1 if f + (0) has no positive eigenvector for an eigenvalue greater than one; (ii) i W (f, P σ ) = 0 if f + (0) possesses a positive eigenvector for an eigenvalue greater than one.
Remark 3. i W (f, P ρ ) denotes the index of f over P ρ with respect to W . More generality, we will denote by i W (T, Z) the index of the operator T over Z with respect to the set Z. Moreover, for an isolated fixed point y of the operator T , i W (T, y) will denote the local index of T at 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]: H(1, y) is an invertible operator on E and the spectral radius of P y D x H(1, y)| My , denoted by Spr (P y D x H(1, y)| My ), is greater than one, then i W (H(1, ·), y) = 0. (ii) If I −D x H(1, y) is an invertible operator on E and Spr (P y D x H(1, y)| My ) < 1, then i W (H(1, ·), y) = (−1) χ , where χ is the sum of the multiplicities of the all eigenvalues of D x H(1, y) greater than one. H(1, y) is an invertible on W y instead of E and there is some w ∈ W y such that the equation (I − D x H(1, y))x = w has no solution x ∈ W y , then i W (H(1, ·), y) = 0.
We use several times the following lemma: Assume that T is a compact and strongly positive linear operator on an ordered Banach space X, with int (P X ) = ∅. Let u > 0 be a positive element of X. We have the following conclusions: (i) If T u > u, then Spr T > 1.
Proof. We will prove the item (i). Assume that u − T u < 0. Since T is a strongly positive linear operator, then T is a positive irreducible operator. Moreover, T is a compact and int (P X ) = ∅. 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, 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 P X is a normal cone and int (P X ) = ∅ because the classical Krein-Rutman Theorem is used (see [1]). However, we are going to use the result for the space C 1 0 (Ω) where the cone is not normal (see [1]).
Proof. (i) From the properties of index, where i P X (H j , N j ) is the index of H j over N j with respect to P X and We will show that i P X (H 1 , N 1 ) = i P X (H 2 , N 2 ) = 1.
For this, we fix M > 0 and we define the homotopies for each (t, u, v) ∈ [0, 1] × N . From the homotopy invariance property, we get
Since γ > σ 1,1 and ρ > σ 1,2 , 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 : P X → P X defined by T u = L 1 (M u+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 T is a compact and strongly positive linear operator.
(v) Observe that once proven (v), (iii) follows by symmetry. Thus, we will prove only (v). We have that By the Maximum Principle of [17], the operator Since ρ > σ 1 (D 2 ; α; F (θ γ (x))K), Proposition 3(iii) implies that v = 0. Hence, On the other hand, from Proposition 4(iii) Thus, by Proposition 3(i) we have that u = 0. Hence, I − D (u,v) H(1, θ γ , 0) is invertible on W y1 . Now, from the Lemma 5.2 it suffices to prove that the operator I − D (u,v) H(1, θ γ , 0) does not have full rank. Suppose that the operator has full rank. Then, take v 0 ∈ P X \ {0} such that L 2 v 0 ∈ P X , then there exists v ∈ P X and f ∈ X satisfying Thus, by Lemma 2.1, we have a contradiction. Therefore, I − D (u,v) H(1, θ γ , 0) does not have full rank, and (v) follows by Lemma 5.2.

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 is less than 1. For this, consider m : Ω −→ IR defined by Observe that the operator T is strongly positive. Indeed, let f ∈ P X and u = T (f ), we get Since M > 0 is sufficiently large, the Maximum Principle of [17] gives us that T (f ) = u ∈ int P X , that is, T is strongly positive. Moreover, T is a compact and linear operator. On the other hand, if ϕ 1 > 0 is an eigenfunction associated to λ 1 (−D 1 ∆ + m(x); γF (θ γ (x))K). Observe that that is, ϕ 1 > T (ϕ 1 ). 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.
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 D 1 and D 2 are both positive. When D 1 and/or D 2 vanish, 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.
Remark 6. We can also study the dependence of the above functions F δ (ρ) and G(γ) with respect to the parameter α. With a similar proof to Proposition 12, we can show: 1. σ 1,2 is a continuous and increasing function on α and σ 1,2 → ∞ as α → ∞.
Hence, for δ = 1, we have the following possible coexistence regions of (1) in the   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.
We will first analyze the case 0 < δ < 1. Observe that E δ → IR 2 as δ → 0 (see Figure 5), hence for any γ > 0 and ρ > 0 there exists δ 0 such that if δ ≤ δ 0 NONLOCAL SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS 1791 ρ σ 1,2 σ 1,1 γ = ℱ 1 (ρ) γ ρ = (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.  . 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 F 1 is above G) and others where the sum is -1 (when G is above F 1 ).

NONLOCAL SYSTEM ARISING FROM THE GROWTH OF CANCER STEM CELLS 1793
As stated above, in general it is not an easy task to ascertain the relative position of the curves γ = F 1 (ρ) and ρ = G(γ) (see [13] and [4] for the classical Lotka-Volterra competition model). In the below lemma we will study a particular case of the relative position of these curves, which ensures that both curves are in the region (γ, ρ) ∈ IR 2 ; ρ ≥ γ .

7.
Conclusion. In this paper we have studied the existence of semi-trivial solutions and coexistence states for a nonlocal elliptic system arising from the growth of cancer stem cells. The model considers the dynamic of cancer stem cells (CSCs) and the non-stem tumor cells (TCs) while are competing for space and resources. In [11] a simplified version (in fact an ode) of this model (the progeny placement depends only on the density at the destination and the density of the cells is uniform) was proposed to investigate the "tumor growth paradox", that means that "an increasing rate of spontaneous cell death in (T Cs) shortens the waiting time for (CSCs) proliferation and migration, and thus facilitates tumor progression". In that paper, the authors show that the unique steady states are (0, 0), (0, v 0 ) (both unstable) and (u 0 , 0) globally stable. Hence, the (TCs) tend to die and the system converges to the pure stem-cell state. Moreover, the authors compare different sizes of the tumor changing α, and they show that the tumor increases as α increases: these results confirm the observations of the tumor growth paradox. As conclusion, they assert that a successful therapy must eradicate cancer stem cells.
In this paper, we consider the general model proposed in [11] (including diffusion, non-uniform population densities and progeny contribution depending on the origin and the destination). We have given results concerning to the existence of semitrivial solutions and coexistence states based on the parameters of the model. From our results, we can conclude: 1. Unlike the simplified model in [11], in our model the coexistence states (both components positive) exist.