Nonlinear Differential Equations, Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 121–133 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

Bifurcation of reaction-diffusion systems related to epidemics ^{∗}

Anthony W. Leung & Beatriz R. Villa

Dedicated to Alan Lazer on his 60th birthday

Abstract

The article considers the reaction-diffusion equations modeling the infection of several interacting kinds of species by many types of bacteria. When the infected species compete significantly among themselves, it is shown by bifurcation method that the infected species will coexist with bacterial populations. The time stability of the postitive steady-states are also considered by semigroup method. If the infected species do not interact, it is shown that positive coexistence states with bacterial populations are still possible.

1 Introduction, Epidemic Models

This article considers reaction-diffusion equations modeling the infection of several interacting kinds of species by many types of virus or bacteria. Rigorous mathematical treatments are presented in the series of papers [1], [2] and [3] for analyzing the infection of one species by one type of bacteria. The situation can become even more involved in reality when there are more than one kind of bacteria and species, which may also interact among themselves. We consider the system:

−Δu_{i} + a_{i}(x)u_{i} = ∑_{j=1}^{m} b_{ij}v_{j} for x ∈ Ω, i = 1, …, n,  (1.1)

−Δv_{k} + ã_{k}(x)v_{k} = ∑_{j=1}^{n} f_{kj}(u_{j}) + v_{k} ∑_{j=1}^{m} c_{kj}v_{j} for x ∈ Ω, k = 1, …, m u_{i} = v_{k} = 0 for x ∈ ∂Ω, i = 1, …, n, k = 1, …, m

where b_{ij} > 0 and c_{kj} ≥ 0 are constants, f_{kj} ∈ C^{1}(R), and Ω is a bounded domain in R^{N}, with ∂Ω of class C^{2+α}, 0 < α < 1. Here, Δ denotes the Laplacian operator; and the corresponding parabolic system, with ∂u_{i}/∂t and ∂v_{k}/∂t added to the first n and the second m equations respectively on the left of (1.1), will also be considered. The functions u_{i} represent n different kinds of bacterial population densities and v_{k} represent m different types of infected species population densities. The populations are assumed to diffuse in space Ω. The functions a_{i}(x) are assumed to be positive, because the bacterial populations tend to die in the absence of other factors; and the terms b_{ij}v_{j} represent the growth of the number bacteria due to infected species. The functions ã_{k}(x) are assumed to be positive, because a certain proportion of the infected species recover per unit time; the terms f_{kj}(u_{j}) describe the rate the k-th species becomes infected by u_{j}, and the terms v_{k}c_{kj}v_{j} describe interaction between the k-th and j-th infected species. The model can be more readily interpreted in the form of the corresponding parabolic system, with (1.1) considered as the steady state solution. The prototype form with m = n = 1, c_{11} = 0, is introduced and explained in e.g. [1] and [2]. We will consider the case when all c_{kj} are zero as well as other cases.

∗Mathematics Subject Classifications: 35B32, 35J60, 35K57, 92D30.

Key words: Reaction-diffusions; Elliptic systems; Parabolic systems; Bifurcations; Epidemiology; Asymptotic stability.

\begin{matrix}c \\ ◯ \end{matrix}2000 Southwest Texas State University.

October 25, 2000.

For convenience, we will adopt the following conventions. Let B and K_{0} be respectively n × m and m × n constant matrices as follows:

B = [ \begin{matrix}b_{11} & ⋯ & b_{1m} \\ ⋮ & & ⋮ \\ b_{n1} & ⋯ & b_{nm} \end{matrix} ] K_{0} = [ \begin{matrix}f′_{11}(0) & ⋯ & f′_{1n}(0) \\ ⋮ & & ⋮ \\ f′_{m1}(0) & ⋯ & f′_{mn}(0) \end{matrix} ] .  (1.2)

Let 𝔈 = {w = col(w_{1}, …, w_{n+m})|w_{i} ∈ C^{1}(Ω̄), w_{i} = 0 on ∂Ω, i = 1, …, m + n}, with norm ∥w∥_{𝔈} = max{∥w_{i}∥_{C^{1}(Ω)}|i = 1, …, m + n}; and 𝔓 denotes the cone 𝔓 = {col.(w_{1}, …, w_{n+m}) ∈ 𝔈|w_{i} ≥ 0 in Ω̄, i = 1, …, n + m}. Also, let 𝔉 = {w = col.(w_{1}, …, w_{n+m})|w_{i} ∈ C^{2+α}(Ω̄), w_{i} = 0 on ∂Ω} with its norm denoted as ∥w∥_{𝔉} = max{∥w_{i}∥_{C^{2+α}(Ω̄)}|i = 1, …, m + n}. As operators from C^{2+α}(Ω) into C^{α}(Ω), we write L_{i} = −Δ + a_{i} for i = 1, …, n, and L_{n+k} = −Δ + ã_{k}, for k = 1, …, m. As an operator from 𝔉 into [C^{α}(Ω̄)]^{n+m}, we write L = col(L_{1}, …, L_{n+m}). For abbreviation, we write F̂ = col(F_{1}, …, F_{n+m}), where F_{j} are operators from [C^{1}(Ω̄)]^{n+m} or 𝔉 into C^{1}(Ω̄) defined by:

F_{i}[col(u_{1}, …, u_{n}, v_{1}, …, v_{m})] = ∑_{j=1}^{m} b_{ij}v_{j} for i = 1, …, n

F_{n+k}[col(u_{1}, …, u_{n}, v_{1}, …, v_{m})] = ∑_{j=1}^{n} f_{kj}(u_{j}) + v_{k} ∑_{j=1}^{m} c_{kj}v_{j} (1.3)
for k = 1, …, m

We now label a few key assumptions, some or all of which will be used in various theorems in this article.

[H1] The functions a_{i} and ã_{k} are members of C^{α}(Ω̄), 0 < α < 1, and satisfy a_{i}(x) > 0, ã_{k}(x) > 0 for all x ∈ Ω̄, i = 1, …, n, k = 1, …, m.

[H2] The functions f_{kj} ∈ C^{1}(R) satisfy f′_{kj}(0) > 0 and f_{kj}(0) = 0, f_{kj}(s) ≥ 0 for all s ≥ 0, k = 1, …, m, j = 1, …, n. For each k, there exists at least one j such that 0 < f_{kj}(s) for all s > 0.

[H2*] In addition to all the properties in [H2], the functions of f_{kj} satisfy f_{kj}(s) ≤ K_{1}s for all s > 0, where K_{1} is some positive constant.

[H3] There exists a constant vector d⃗ = col(d_{1}, …, d_{n}), with d_{i} > 0, i = 1, …, n such that: BK_{0}d⃗ > (λ_{1} + a^{∗})^{2}d⃗.

In [H3] and the rest of the paper, the (strict) inequality between the two vectors is interpreted to be satisfied for each component. The quantity λ_{1} denotes the first eigenvalue of the problem: −Δϕ = λϕ in Ω, ϕ = 0 on ∂Ω; and ϕ will denote the corresponding positive eigenfunction with ∥ϕ∥_{∞} = 1. The symbols a^{∗} and a^{∗∗} are defined as a^{∗} = sup{a_{i}(x), ã_{k}(x)|x ∈ Ω̄, i = 1, …, n, k = 1, …, m} and a^{∗∗} = inf{a_{i}(x), ã_{k}(x)|x ∈ Ω̄, i = 1, …, n, k = 1, …, m}. Another assumption which will sometimes be used concerning the interaction of the species v_{k} is as follows:

[H4] c_{kk} < 0 and |c_{kk}| > ∑_{j=1,j≠k}^{m} |c_{kj}|, for each k = 1, …, m.

In Section 2, we will show that problem (1.1) has a positive solution under appropriate conditions. Essentially, we assume that intraspecies interaction among infected species is large compared with interspecies interactions (cf. hypothesis [H4]). In Section 3 we consider the time stability of the positive solution as a steady-state of the corresponding parabolic system, when the infected species v_{k} compete among themselves (i.e. c_{kj} < 0). In Section 4 we consider the situation when the infected species v_{k} do not interact (i.e. c_{kj} = 0). We show that a positive solution still exists, if we assume appropriate infection rates when large numbers of bacteria are present (cf. [H5]). We will use bifurcation methods to show the existence of positive steady-state solutions, and semigroup method to analyze stability of steady-states. In many cases, we only outline the ideas of the proofs, the complete details will be shown elsewhere.

2 Bifurcation of Infected Species

In this section, we will show that under hypotheses [H1], [H2*], [H3] and [H4], the problem (1.1) has a positive solution. The main result of this section is Theorem 2.2. Let w = col(u_{1}, …, u_{n}, v_{1}, …, v_{m}), the system (1.1) can be abbreviated as

L[w] = F [w], where w ∈ 𝔉,  (2.1)

with the operators L and F as introduced before. To study this nonlinear problem, we consider the auxiliary problem:

L[w] = λF [w], w ∈ 𝔉,  (2.2)

where λ is a parameter, and investigate the bifurcation from the trivial solution w = 0 as the parameter λ passes through a certain value λ_{0}. Under conditions [H1] and [H2], we will see that this bifurcation actually occurs in Theorem 2.1. Moreover, in Theorem 2.2, we will see that hypotheses [H2*] and [H3] insure that λ_{0} < 1; and hypothesis [H4] insures that the bifurcation curve of nontrivial solutions connects to λ = +∞. Thus (2.2) has a nontrivial solution when λ = 1, i.e. (2.1) has a nontrivial solution under appropriate conditions. For convenience, we let M_{0} denotes the (n + m) × (n + m) square constant matrix:

M_{0} = [ \begin{matrix}0 & B \\ K_{0} & 0 \end{matrix} ]

where the 0’s along the diagonal are zero matrices with appropriate dimensions. Applying L^{−1} to both sides of (2.2), using zero Dirichlet boundary condition, we obtain: w = λL^{−1}F [w]. Thus, (2.2) can be written as:

Q(λ, w) = 0, (λ, w) ∈ R × 𝔈,  (2.3)

where Q : R × 𝔈 → 𝔈 is an operator given by

Q(λ, w) := w − λL^{−1}F [w]

(for the entire paper the inverse operators L^{−1} or L_{i}^{−1} will always mean finding the solution using zero Dirichlet boundary condition).

Theorem 2.1 Under hypotheses [H1] and [H2], the point (λ_{0}, 0) is a bifurcation point for problem (2.3). Here λ = λ_{0} is the unique positive number so that the problem:

L[w] = λM_{0}w in Ω, w = 0 on ∂Ω,  (2.4)

has a nonnegative eigenfunction in 𝔈. (The eigenvalue λ_{0} is simple.) Moreover, the component of 𝔖̄ containing the point (λ_{0}, 0) is unbounded, where

𝔖 := {(λ, w) ∈ R^{+} × 𝔓|Q(λ, w) = 0, λ > 0 and w ∈ 𝔓\{0}};

and it also has the property that 𝔖̄ ∩ (R × ∂𝔓) = (λ_{0}, 0).

We first state a sequence of Lemmas which will lead to the proof of Theorem 2.1.

Lemma 2.1 (Comparison). Let w, ŵ ∈ [C^{2}(Ω) ∩ C^{1}(Ω̄)]^{n+m}, w ≢ 0, ŵ_{i} ≥ 0, w_{i} ≢ 0 in Ω, for i = 1, …, n + m, satisfy

L_{i}[w_{i}(x)] = ∑_{j=1}^{n+m} p_{ij}(x)w_{j}(x), for x ∈ Ω, i = 1, …, n + m

w|_{∂Ω} = 0, w = col(w_{1}, …, w_{n+m}),

L_{i}[ŵ_{i}(x)] ≥ ∑_{j=1}^{n+m} q_{ij}(x)ŵ_{j}(x), for x ∈ Ω, i = 1, …, n + m

ŵ = col(ŵ_{1}, …, ŵ_{n+m})

where p_{ij} and q_{ij} are bounded functions in Ω. Suppose that

q_{ij} ≥ p_{ij} in Ω̄ for i, j = 1, …, n + m,
and q_{ij}, p_{ij} ≥ 0 in Ω̄ for all i ≠ j,

then there exists an integer k, 1 ≤ k ≤ n + m and a real number δ such that

ŵ_{k} ≡ δw_{k}, p_{kj} ≡ q_{kj} in Ω̄ for all j = 1, …, n + m
and ŵ_{j} − δw_{j} ≥ 0 for all j = 1, …, n + m.

This lemma is exactly the same as Lemma 2.1 in [10], the proof will thus be omitted.

Lemma 2.2 Under hypotheses [H1] and [H2], there exists (λ_{0}, w^{0}) ∈ R × 𝔉, λ_{0} > 0, such that

L[w^{0}] = λ_{0}M_{0}w^{0} in Ω, w^{0} = 0 on ∂Ω  (2.5)

with each component w_{i}^{0} > 0 in Ω, ∂w_{i}^{0}/∂ν < 0 on ∂Ω for i = 1, …, n+m. Furthermore, 1/λ_{0} is a simple eigenvalue of the operator L^{−1}M_{0} : [C^{1}(Ω̄)]^{n+m} → [C^{1}(Ω̄)]^{n+m} (that is the eigenfunction corresponding to this eigenvalue is unique up to a multiple). The number λ = λ_{0} is the unique positive number so that the problem w = λL^{−1}M_{0}w has a nontrivial nonnegative solution for w ∈ 𝔓.

Lemma 2.3 Let G : 𝔈 → 𝔈 be the operator defined by:

G[w] = L^{−1}[F (w) − M_{0}w],

then ∥G[w]∥_{𝔈}/∥w∥_{𝔈} → 0 as ∥w∥_{𝔈} → 0.

The details for proving these lemmas will be shown in a forthcoming article.

Outline of Proof of Theorem 2.1: The operator G described above is completely continuous; and the operator L^{−1}M_{0} described in Lemma 2.2 is compact and positive with respect to 𝔓. Equation (2.3) can be written as:

w − λL^{−1}[M_{0}w] − λG[w] = 0, for (λ, w) ∈ R^{+} × 𝔈.

By means of Lemma 2.3 and the existence and uniqueness part of Lemma 2.2 we can apply Theorem 29.2 in [5] to conclude that (λ_{0}, 0) is a bifurcation point for problem (2.3), and the component of 𝔖̄ containing the point (λ_{0}, 0) as described above is unbounded.

Let (λ_{i}, w_{i}) ∈ 𝔖, i = 1, 2, … be a sequence tending to a limit point (λ̄, w̄) in R × ∂𝔓. Using the limiting equation for (λ̄, w̄), we can show by means of the maximum principle and the uniqueness part of Lemma 2.2 that (λ̄, w̄) = (λ_{0}, 0). The details will be shown in a later article.

Using additional hypotheses, we can show by means of Theorem 2.1 that the original problem (2.1) or (1.1) has a nonnegative solution.

Theorem 2.2 Under hypotheses [H1], [H2*], [H3] and [H4], the problem (2.1) has a solution w = col(w_{1}, …, w_{n+m}) ∈ 𝔉, such that w_{i} ≥ 0 in Ω for each i and w ≢ 0 (i.e. w ∈ 𝔓\{0}).

In order to prove Theorem 2.2, we first show the following two lemmas.

Lemma 2.4 Under hypotheses [H1], [H2] and [H3], the positive number where bifurcation occurs described in Theorem 2.1 satisfies λ_{0} < 1.

To prove this lemma, we first use [H3] to find a constant vector g⃗ such that L^{−1}[M_{0}g⃗ϕ] ≥ rϕg⃗ in Ω for some r > 1. Then we use Theorem 2.5 in [7] to obtain λ_{0} ≤ \frac{1}{r} < 1.

Lemma 2.5 Under the hypotheses [H1] and [H2], let (λ̄, w̄) ∈ 𝔖, where 𝔖 is described in Theorem 2.1. Suppose R_{k} are positive constants such that 0 ≤ w̄_{n+k}(x) ≤ R_{k} for all x ∈ Ω̄, k = 1, …, m. Then for each i = 1, …, n

0 ≤ w_{i}(x) ≤ λ̄(inf a_{i})^{−1} ∑_{k=1}^{m} b_{ik}R_{k} for all x ∈ Ω̄.

This lemma is proved by the method of upper-lower solution.

Outline of Proof of Theorem 2.2: Let the component of 𝔖̄ containing the point (λ_{0}, 0), described in Theorem 2.1, be denoted by 𝔖^{+}. Since λ_{0} < 1, by Lemma 2.4, it suffices to show that the set I := {λ ∈ R^{+}|(λ, w) ∈ 𝔖^{+} for some w} is unbounded.

We can show by means of [H4] and [H2*] that if there exists λ such that (λ, w) ∈ 𝔖^{+}, then w = col(w_{1}, …, w_{n+m}) must satisfy:

0 ≤ w_{i}(x) ≤ N̂ for all x ∈ Ω̄, i = n + 1, …, n + m.  (2.6)

for some positive constant N̂.

Finally, inequality (2.6), Lemma 2.5 and gradient estimates by means of equation (2.2) imply that 𝔖^{+} cannot be unbounded if I is bounded. Consequently, I must be unbounded, and this completes the proof of the Theorem 2.2.

3 Stability of Infected Competing Species

In this section, we will consider the stability of the steady-state solutions found in the last section as a solution of the corresponding parabolic system. It will be seen in Theorem 3.2 that if [H2] and [H4] are strengthened, then the bifurcating steady-states near the bifurcation point are asymptotically stable in time. Before obtaining further results with additional hypotheses, we first deduce a few more consequences of hypotheses [H1] and [H2].

Lemma 3.1 Under hypotheses [H1] and [H2], the problem

L[w] = λM_{0}^{T} w in Ω, w = 0 on ∂Ω,  (3.1)

has a solution (λ, w) = (λ_{0}, ŵ^{0}), ŵ^{0} ∈ 𝔉, with each component ŵ_{i}^{0} > 0 in Ω, ∂ŵ_{i}^{0}/∂ν < 0 on ∂Ω for i = 1, …, n + m. (Here, λ_{0} is exactly the same number as in Lemma 2.2.) Moreover, any solution of (3.1) with λ = λ_{0} is a multiple of ŵ^{0}.

Proof. The existence of a positive solution and the simplicity of the corresponding eigenvalue is proved in exactly the same way as Lemma 2.2 with the role of B and K_{0} interchanged. The fact that λ_{0} is exactly the same as in Lemma 2.2 follows exactly the same procedure as in the proof of Lemma 2.4 in [10], and will thus be omitted.

For convenience, we will define two operators L_{0} and L_{1} : 𝔈 → 𝔈 as follows:

L_{0} := I − λ_{0}L^{−1}M_{0}  (3.2)
L_{1} := −L^{−1}M_{0}  (3.3)

Lemma 3.2 Under hypotheses [H1] and [H2], the null space and range of L_{0}, denoted respectively by N(L_{0}) and R(L_{0}) satisfy:

(i) N(L_{0}) is one-dimensional, spanned by w^{0};

(ii) dim[𝔈/R(L_{0})] = 1;

(iii) L_{1}w ∉ R(L_{0}).

Proof. Part (i) was proved in Lemma 2.2. The remaining parts are proved in the same way as in Lemma 2.5 in [10]. For the proof of part (iii), the positivity property of w^{0} and ŵ^{0} is used.

Theorem 3.1 Assume hypotheses [H1], [H2] and the additional condition that f_{kj} ∈ C^{2}(R) for k = 1, …, n, j = 1, …, m. Then there exists δ > 0 and a C^{1}-curve (λ̂(s), ϕ̂(s)) : (−δ, δ) → R × 𝔈 with λ̂(0) = λ_{0}, ϕ̂(0) = 0 such that in a neighborhood of (λ_{0}, 0), any solution of (2.3) is either of the form (λ, 0) or on the curve (λ̂(s), s[w^{0} + ϕ̂(s)]) for |s| < δ. Furthermore, the set 𝔖^{+}\{(λ_{0}, 0)} is contained in R^{+} × (Int 𝔓), where 𝔖^{+} is the component of the closure of 𝔖 (described in Theorem 2.1) containing the point (λ_{0}, 0) in R^{+} × 𝔈.

Outline of Proof. Equation (2.3) can be written as

Q(λ, w) := L_{0}[w] + (λ − λ_{0})L_{1}[w] − λG[w] = 0  (3.4)

where the operator G : 𝔈 → 𝔈 is defined in Lemma 2.3. Under the additional smoothness condition on f_{jk}, we can show that Q ∈ C^{2}(R^{+} × 𝔈, 𝔈) and the Frechet derivative of G is continuous on 𝔈. Moreover, we can readily deduce as in Lemma 2.3 that L_{0} = D_{2}Q(λ_{0}, 0), L_{1} = D_{12}Q(λ_{0}, 0), and G[0] = D_{2}G[0] = 0. Hence we can apply the local bifurcation theorem in [4] to obtain the C^{1}- curve (λ̂(s), ϕ̂(s)) describing the nontrivial solution of (2.3) near (λ_{0}, 0) as stated above. The remaining details will be shown elsewhere.

In the remaining part of this section, we will consider the linearized and asymptotic stability of the positive bifurcating solution described in Theorem 3.1, near (λ_{0}, 0). Applying the bifurcation theory in [4], and the fact that

∫_{Ω} ŵ^{0} · Δ^{−1}w^{0}dx ≠ 0  (3.5)

(note that each component of both ŵ^{0} and w^{0} is strictly positive in Ω), we can assert that there exist δ_{1} ∈ (0, δ) and two functions:

(γ(·), z(·)) : (λ_{0} − δ_{1}, λ_{0} + δ_{1}) → R × 𝔈,
(η(·), h(·)) : [0, δ_{1}) → R × 𝔈,

with (γ(λ_{0}), z(λ_{0})) = (η(0), h(0)) = (0, w^{0}), such that

D_{2}Q(λ, 0)z(λ) = γ(λ)(−L)^{−1}(z(λ)), and  (3.6)
D_{2}Q(λ̂(s), s(w^{0} + ϕ̂(s))h(s) = η(s)(−L)^{−1}(h(s))  (3.7)

Here (3.5) and the theory in [4] imply that γ(λ) and η(s) are respectively L^{−1}- simple eigenvalues of D_{2}Q(λ, 0) and D_{2}Q(λ̂(s), s(w^{0}+ϕ̂(s)), with eigenfunctions z(λ) and h(s). Moreover, the theory in [4] further leads to the following lemmas.

Lemma 3.3 Assume all the hypotheses in Theorem 3.1. There exists ρ > 0 such that for each s ∈ [0, δ_{1}), there is a unique (real) eigenvalue η(s) for the linear operator

Q_{s}^{∗} := −LD_{2}Q(λ̂(s), s(w^{0} + ϕ̂(s)) : 𝔉 → [C^{α}(Ω̄)]^{m+n}  (3.8)

satisfying |η(s)| < ρ with eigenfunction h(s) ∈ 𝔉. That is,

Q_{s}^{∗}h(s) ≡ −L[h(s)] + λ̂(s)F_{w}[s(w^{0} + ϕ̂(s))]h(s) = η(s)h(s)  (3.9)

The next few lemmas study the behavior of the eigenvalues λ̂(s), η(s) for small s ≥ 0, and γ(λ) near λ = λ_{0}.

Lemma 3.4 Assume all the hypotheses of Theorem 3.1. Suppose further

[H2** ] f′′_{kj}(0) < 0 for k = 1, …, m, j = 1, …, n;

[H4* ] c_{kj} < 0, for all k, j = 1, …, m.

Then the function λ̂(s) satisfies λ̂′(0) > 0.

Lemma 3.5 Under all the hypotheses in Theorem 3.1, the function γ(λ) satisfies γ′(λ_{0}) > 0.

Lemma 3.6 Under the hypotheses of Theorem 3.1, [H2^{∗∗}] and [H4^{∗}], there exists δ_{2} ∈ (0, δ_{1}) such that η(s) < 0 for all s ∈ (0, δ_{2}).

The proof of the last three lemmas will be shown elsewhere.

Equation (2.2) or (2.3) can be written as

−L[w] + λF [w] = 0  (3.10)

The linearized eigenvalue problem for (3.10) at the bifurcating solution w = s(w^{0} + ϕ̂(s)) is precisely (3.9). When λ = λ̂(0) = λ_{0}, the eigenvalue problem corresponding to (3.9) becomes

−L[h] + λ_{0}M_{0}h = ηh  h ∈ 𝔈  (3.11)

where η is the eigenvalue. Under hypotheses [H1] and [H2], Theorem 2.1 asserts that η = 0 is an eigenvalue for (3.11), with positive eigenfunction. Using this property and the fact that the off diagonal terms of M_{0} are all nonnegative, we can proceed to show the following.

Lemma 3.7 Under hypotheses [H1] and [H2], all eigenvalues in equation (3.11) except η = 0 satisfies Re(η) < −r for some positive number r.

Lemma 3.4 to 3.6 above essentially shows the eigenvalue η = 0 corresponding to (3.9) at s = 0 moves to the left as s increases. As to the other eigenvalues with Re(η) < −r described in Lemma 3.7, one can show by perturbation arguments that they should still be in the left open half plane for s > 0 sufficiently small. More precisely, we have the following.

Lemma 3.8 Under the hypotheses of Theorem 3.1, there exists a number δ^{∗} ∈ (0, δ) and a positive function η(s) for s ∈ (0, δ^{∗}) such that the real parts of all the numbers in the point spectrum of the linear operator Q_{s}^{∗} are contained in the interval (−∞, −η(s)), for s ∈ (0, δ^{∗}). (Here, δ is described in Theorem 3.1 and Q_{s}^{∗} is described in (3.8) in Lemma 3.3).

For each s ∈ (0, δ^{∗}), the function w̄_{s} := s[w^{0} + ϕ̂(s)] described in Theorem 3.1 can be considered as a steady-state solution of the problem:

\frac{∂w}{∂t} (x, t) + L[w(x, t)] = λF [w(x, t)] in Ω × (0, ∞)  (3.12)
w(x, t) = 0 x ∈ ∂Ω, t ≥ 0 .

We now consider the time asymptotic stability of this steady-state as a solution of the parabolic system (3.12). In order to obtain a precise statement, we let B_{1} and B_{2} be Banach spaces as follow:

B_{1} = {u : u ∈ [C(Ω̄)]^{n+m}, u = 0 on ∂Ω} and
B_{2} = {u : u ∈ [L_{p}(Ω)]^{n+m}} for p large enough such that N/(2p) < 1.

Let A_{1} be the operator L on B_{1} with domain D(A_{1}) = {u : u ∈ [W^{2,p}(Ω)]^{n+m} for all p, Δu ∈ [C(Ω̄)]^{n+m}, u = 0 and Δu = 0 on ∂Ω}; and A_{2} be the operator L on B_{2} with domain D(A_{2}) = {u ∈ B_{2} : u ∈ [W^{2,p}(Ω) ∩ W_{0}^{1,p}(Ω)]^{n+m}}. For w = col(w_{1}, …, w_{n+m}), we consider the following nonlinear initial-boundary value problem for each i = 1, 2 corresponding to (3.12):

\frac{dw}{dt} + A_{i}w(t) = λF (w(t)), w(0) = w_{0} for t ∈ (0, T]  (3.13)

A solution of (3.13) in B_{i} is a function w ∈ C([0, T], B_{i}) ∩ C^{1}((0, T], B_{i}), with w(0) = w_{0}, w(t) ∈ D(A_{i}) for all t ∈ (0, T]; and w(t) satisfies (3.13) for t ∈ (0, T]. The operator A_{2} is an infinitesimal generator of an analytic semigroup, say M(t), on B_{2} for t ≥ 0. It is well known that for α > 0

(−A_{2})^{−α} = \frac{1}{Γ(α)} ∫_{0}^{∞} τ^{α−1}M(τ)dτ

defines a bounded linear operator on B_{2}. Moreover, [(−A_{2})^{−α}]^{−1} = (−A_{2})^{α} is a closed operator on B_{2} with dense domain D((−A_{2})^{α}) = (−A_{2})^{−α}(B_{2}). We denote by X_{α} the Banach space (D(−A_{2})^{α}, ∥ ∥_{α}) where ∥w∥_{α} = ∥(−A_{2})^{α}w∥_{L^{p}} for all w ∈ D((−A_{2})^{α}).

Using Lemma 3.8, we can use semigroup theory and apply the stability Theorem 5.1.1 in [6] for sectorial operators to obtain the following asymptotic stability theorem for the steady-state solution w̄_{s}. The details of the proof are omitted here.

Theorem 3.2 Assume all the hypotheses of Theorem 3.1, [H2^{∗∗}] and [H4^{∗}]. For each fixed s ∈ (0, δ^{∗}), let λ = λ̂(s), w̄ = s[w^{0} + ϕ̂(s)]. Then for each i = 1, 2, there exists ρ > 0, β > 0 and M > 1 such that equation (3.13) has a unique solution in B_{i} for all t > 0 if w_{0} ∈ B_{1} and ∥w_{0} − w̄∥_{∞} ≤ ρ/(2M) for i = 1, (or w_{0} ∈ X_{α} and ∥w_{0} −w̄∥_{α} ≤ ρ/(2M) for i = 2.) Moreover, the solution satisfies

∥w(t) − w̄∥_{∞} ≤ 2Me^{−βt}∥w_{0} − w̄∥_{∞} for all t ≥ 0, i = 1, or  (3.14)
∥w(t) − w̄∥_{α} ≤ 2Me^{−βt}∥w_{0} − w̄∥_{α} for all t ≥ 0, i = 2.  (3.15)

(For solutions in B_{2}, we assume α ∈ (N/(2p), 1) for the space X_{α}.)

4 Bifurcation of Infected Species with No Interactions

In this section, we consider system (1.1), under the special situation when all c_{kj} = 0, k, j = 1, …, m; that is, we consider the problem:

−Δu_{i} + a_{i}(x)u_{i} = ∑_{j=1}^{m} b_{ij}v_{j} for x ∈ Ω, i = 1, …, n
−Δv_{k} + ã_{k}(x)v_{k} = ∑_{j=1}^{n} f_{kj}(u_{j}) for x ∈ Ω, k = 1, …, m  (4.1)
u_{i} = v_{k} = 0 for x ∈ ∂Ω, i = 1, …, n, k = 1, …, m .

In other words, the infected species v_{k} will not interact among themselves. This situation is a direct generalization of the theory in [1]. Under additional assumptions on f_{kj}(u_{j}) for large u_{j} (see [H5] below), Theorem 4.1 shows that problem (4.1) has a positive solution. Letting w = col(w_{1}, …, w_{n+m}) = col(u_{1}, …, u_{n}, v_{1}, …, v_{m}), system (4.1) can be written as:

L[w] = F̃[w], w ∈ 𝔈,  (4.2)

where F̃ is the same as F described in (1.3) with the special restriction c_{kj} = 0, for all k, j = 1, …, m. For convenience, we define

f̂_{ij}(η) := \{ \begin{matrix}f_{ij}(η)/η & if & η ≠ 0 \\ f′_{ij}(0) & if & η = 0 \end{matrix} for i = 1, …, m, j = 1, …, n. (4.3)

Also define the m × n matrix

[f̂_{ij}(η_{ij})]_{ij=1}^{m,n} := [ \begin{matrix}f̂_{11}(η_{11}) & ⋯ & f̂_{1n}(η_{1n}) \\ ⋮ & & ⋮ \\ f̂_{m1}(η_{m1}) & ⋯ & f̂_{mn}(η_{mn}) \end{matrix} ]  (4.4)

where η_{ij} are real numbers for i = 1, …, m, j = 1, …, n. We will use the following hypothesis:

[H5 ] There exist a real number η_{0} > 0, and a constant vector q⃗ = col(q_{1}, …, q_{n}), with q_{i} > 0, i = 1, …, n, such that:

q^{T} B [f̂_{ij}(η_{ij})]_{i,j=1}^{m,n} < (λ_{1} + a^{∗∗})^{2}q^{T} for all η_{ij} ≥ η_{0}

Under hypothesis [H5], one can always choose a number ρ_{1} with 0 < ρ_{1} < (λ_{1} + a^{∗∗}) such that:

q^{T} B [f̂_{ij}(η_{ij})]_{i,j=1}^{m,n} < ρ_{1}^{2}q^{T} < (λ_{1} + a^{∗∗})^{2}q^{T} for all η_{ij} ≥ η_{0}  (4.5)

The following theorem is the main result of this section.

Theorem 4.1 Under hypotheses [H1], [H2^{∗}], [H3] and [H5], the problem (4.2) (alternatively, problem (4.1) with w = col(u_{1}, …, u_{n}, v_{1}, …, v_{m})) has a solution w = col(w_{1}, …, w_{n+m}) ∈ 𝔉, such that w_{i} ≥ 0 in Ω̄ for each i and w ≢ 0 (i.e. w ∈ 𝔓\{0}).

In this entire section, F is considered with the special restriction c_{kj} = 0, and [H4] is not assumed. We are thus led to the problem:

L[w] = λF̃[w] w ∈ 𝔈  (4.6)

with F̃ as described above. Since F̃ is a special case of F, Therem 2.1 applies. Under the assumptions of Theorem 4.1, let 𝔖 be as defined in Theorem 2.1, and 𝔖^{+} be the component of 𝔖̄ containing the point (λ_{0}, 0). Recall that 𝔖^{+} is proved to be unbounded in Theorem 2.1. The following Lemma will be needed in the proof of Theorem 4.1.

Lemma 4.1 Assume all the hypotheses of Theorem 4.1. Suppose {(λ̃_{r}, w̃^{r})}, r = 1, 2, … is a sequence in 𝔖^{+} with the property: λ̃_{r} → λ̂, 0 < λ̂ < ∞, and ∥w̃^{r}∥_{𝔈} → ∞, as r → ∞. Then there exists a subsequence {(λ̃_{r(j)}, w̃^{r(j)})} such that the first n components of w̃^{r(j)} tend to +∞ uniformly in compact subsets of Ω, as r(j) → ∞.

Outline of Proof of Theorem 4.1: Recall that Lemma 2.4 implies that λ_{0} < 1. To prove this Theorem, it suffices to show the fact that if there exists a sequence in 𝔖^{+} with property as described in Lemma 4.1, then we must have λ̂ > 1. To show this fact, we use hypothesis [H5] concerning the functions f_{kj}(u_{j}) for large u_{j}. It will lead to λ̂ ≥ (λ_{1} + a^{∗∗})/ρ_{1} > 1. This leads to the existence of (λ̃, w̃) in 𝔖^{+} with λ̃ = 1; that is, we obtain a nontrivial, nonnegative solution of Problem (4.1).

Acknowledgement: The authors would like to express their appreciation to Professor Luis Ortega for his valuable discussion on the subject.

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Anthony W. Leung Department of Mathematical Sciences University of Cincinnati Cincinnati OH 45221-0025, USA e-mail: Anthony.Leung@uc.edu

Beatriz R. Villa Department of Mathematics Universidad Nacional de Colombia Bogota, Colombia e-mail: bvilla@matematicas.unal.edu.co