Monotone traveling waves in a general discrete model for populations

In this paper we consider the existence of monotone traveling waves for a class of general integral difference model for populations that allows the dispersal probability to have no continuous density functions but the fecundity functions to generate a monotone dynamical systems. In this setting we deal with the non-compactness of the evolution operator by using the monotone iteration method.

1. Introduction. In [18] the growth of plant populations with seed bank is studied with the model N n+1 = (1 − γ)g(N n ) + γρN n , where N n is represents the mature plant population at time n, g(N n ) stands for the seed production, the fraction of seeds that survive the dormant period, the fraction of these surviving seeds that will germinate, and the fraction of seedlings that will survive to yield a mature plant; γ and ρ are the fraction of seeds in the seed bank that do not germinate, and the fraction of non-germinating seeds that survive for a whole year, respectively. The term (1 − γ)g(N n ) describes the growth from new seeds, while the term γρN n stand for the growth due to old seeds in the seed bank.
In [8] the author considers the above populations with space factor that is included to study the existence of monotone traveling waves among other things. To this purpose, the following model is introduced k(x − y)g(u n (y))dy + γρu n (x), (1.1) where u n (x) represents the density of the mature plant population at time n, and k is seed dispersal kernel. In this model it is understood that k(x − y) is the density function of the probability P (x, y) of an individual to migrate from location y to location x in the habitat.
In the well known theory of traveling waves for integro-difference models for the migration of populations (see [30]) if γ = 0, and with a compactness condition that 3222 THUC MANH LE AND NGUYEN VAN MINH follows from the continuity of the kernel k traveling waves exist. The case when γ = 0 gives rise to a challenge when the compactness is absent. However, when k is assumed to be continuous a "quasi-compactness" is still available for a modification of the well known method (developed in [11]) to prove the existence of (possibly non-monotone) traveling waves (see [8]). We note that certain non-compactness cases is also treated in [17,29] with different methods.
It is our purpose in this paper to consider the existence of monotone traveling waves for some models with more general form of probability P (x, y) for the case of monotone fecundity function. Namely, we may assume that the migration probability P (x, y) may have no density function, and if there is, the density k(·) may not be continuous, but just piecewise continuous, and the fecundity function F (x) is the Ricker recruitment function F (x) = xe r−x , where 0 < r ≤ 1. The non-compactness difficulty arising because of this will be overcome by exploiting as much as possible the monotonicity of the dynamical system. We will use the monotone iteration method coupled with the general theory of spreading speeds and traveling waves developed by Weinberger in [30]. Our main result is stated in Theorem 2.8.
That being said, the model (1.1) may be generalized to take the form where we assume that P (x − y) is the probability of an individual to migrate from a location y to location x in the habitat, that depends only on the distance of x − y. A reasonable model with such a general form of the probability P is the following where K is a non-negative and measurable function. With this in mind we will consider the following model (1.4) where N n (x) is the population densities in year n, and ρ > 0; η i , s ≥ 0 such that c i , n = 1, 2, · · · , L, are any fixed locations of the habitat of R. Before we move forward with a study of the general model (1.4) let us consider some special cases when c i = 0 for all i = 1, 2, · · · . In this case we will set (1.6) Eq. (1.6) with s = 0 is a partially sedentary model for population that is studied in ( [28], [29]), while Eq. (1.6) with η = 0 is a model for plant population with seed bank (see [8]). Therefore, Eq. (1.6) could be seen as a combination of the models in [29] and [8].
In most of previous works (see e.g. [8,29] and their references) P (x, y) is assumed to have continuous density function, that is, η i = 0, i = 1, 2, ...L, and K is continuous. This assumption seems to be technical in order to use a compactness (or quasi-compactness) of the evolution operator Q in the proof of existence of traveling waves.
Notations and assumptions. We denote by N, Z, and R the set of natural numbers, set of integers, and set of the reals, respectively. We also denote by BM (R, R) (BC(R, R), respectively) the space of all measurable and bounded real valued functions on R (the space of all bounded continuous real valued functions on R, respectively) with sup-norm. For a constant α we will denote the constant function 2. Main results. In this paper we will assume that the density K(·) satisfies the following properties: iii) For all x 0 ∈ R, the following is true Definition 2.1. A function defined on the real line is said to be "piecewise continuous on the real line" if there is a sequence of ascending numbers T k , ∈ N, or, Z such that it is continuous on every open interval (T k , T k+1 ) and has finite limits at each endpoint. Proposition 2.2. Assume that K satisfies Conditions i) and ii) in the above assumption. Then, Condition iii) in the Standing Assumption is satisfied if the function K(·) is bounded and piecewise continuous on the real line.
Proof. Assume that K is essentially bounded by a positive M on the real line. Then, for every given > 0 there exists a positive T such that Note that there are only finitely many points of discontinuity of K in the interval Finally, if in addition to other choices for δ we can choose δ so small that 2N δ < /4, we can make Combined this estimate with (2.2) we get (2.1). The proposition is proved.
In the rest of this paper we assume that the function F is the Ricker recruitment function F (x) = xe r−x , where 0 < r ≤ 1, although it could be a monotone and non-negative function with similar properties as in our later discussion.
Let us define a dynamical system N n+1 = Q[N n ] by setting where N ∈ BM (R, R). Let us consider the function F (x) := xe r0−x where 0 < r 0 ≤ 1. Then F (0) =) and F (r 0 ) = r 0 . Let us denote It is easy to see that the function has two fixed points 0 and From now on we will assume By assumption (1.5), we have ln((1 − s)/θ) ≥ 0, so 0 < r ≤ r 0 ≤ 1. On the interval [0, r], the function g(u) := θF (u) + su is increasing, and has two fixed points u = 0, u = r. The following lemma can be proved so that the general theory for the spreading speed [30] can be applied.
Lemma 2.3. Under the above notations and assumption on F , if K satisfies the Standing Assumption and Condition (2.1), then the following assertions are true: i) Q is an operator acting in BM (R, R) leaving BC(R, R) and B r invariant; moreover, Q maps an increasing (non-increasing, respectively) function to an increasing (non-increasing, respectively) function ii) Proof. i) Obviously, Q acts in BM (R, R). To show that this operator leaves BC(R, R) invariant we notice that it is equivalent to show that the integral ∞ −∞ K(x − y)F (N (y))dy depends continuously on x if N is a continuous and bounded function. In fact, for a fixed x 0 we have Since This yields the continuity of the function Q[N ], or, the operator Q leaves BC(R, R) invariant. For the invariance of B r it follows immediately from the formula defining Q and the fact that for x ∈ [0, r] we have g(x) ∈ [0, r], so g(u(t)) ∈ [0, r] for all u ∈ B r .
Let N (·) be an increasing function. Then, to show that In fact, since the function F (x) = xe r−x that is increasing when 0 < r < 1 Similarly we can show that if N is non-increasing the function Q[N ] is also nonincreasing.
ii) Obviously, Q[0] = 0, Q[r] = r. Since 0 < α < r, and F (x) = xe r−x with 0 < r ≤ 1, for the constant function α as an element in B(R, R) we have F (α) > α. As shown in [30] the sequence {a n (c; ·)} is non-increasing and bounded, so for each s ∈ R, we obtain the pointwise limit lim n→∞ a n (c; s) = a(c; s), s ∈ R. (2.5) Obviously, 0 ≤ a(c; s) ≤ r for all s ∈ R. The following number is called spreading speed for our model c * := sup{c|a(c; +∞) = r}. (2.6) Similarly we will define a number c * − as follows: and a sequence of functions {b n (c; ·)} by Again, as shown in [30] the sequence {b n (c; ·)} is non-increasing and bounded, so for each s ∈ R, we obtain the pointwise limit By the theory in [30], if there exists a bounded non-negative measure m(x, dx) on R such that (2.14) As F (x) = xe r0−x with 0 < r 0 ≤ 1 we have F (x) ≤ xe r0 for all x ∈ [0, r 0 ]. Therefore, for each u ∈ C r , To proceed we define where δ(x) is the Dirac's delta function, that is, for all a ∈ R

THUC MANH LE AND NGUYEN VAN MINH
By the theory of spreading speed in [30], we have Similarly, Substituting w(x − nc) into N n (x) in (1.4) gives Let us make a change of variables ξ := x − (n + 1)c to re-write the above equation in the following form: For every fixed constant c ∈ R we will denote We will write this operator Q c as the sum of two operators as follows We note that for sufficiently small s, η i the operator A c is a contraction. However, the operator B c is not compact in the space of non-increasing continuos functions w satisfying the boundary conditions lim x→∞ w(x) = 0, lim x→−∞ w(x) = r due to the potential discontinuity of the kernel K. For this reason we will not be able to use the fixed point theorem for compact operators in a closed and convex space to prove the existence of fixed points to the operator Q c . Instead we will make use of the monotone iteration method to construct a monotone sequence of functions that is convergent to a monotone traveling wave solution. Proof. To show the existence of the operator (I − A c ) −1 in BM (R, R) we will solve w ∈ BM (R, R) from the equation given u ∈ BM (R, R). Set S u (w) = A c (w) + u and notice that S u is a strictly contractive because By (2.26) this yields that the operator S u is a strict contraction in BM (R, R), so by the Contraction Mapping Principle it has a unique fixed point, say w. This fixed point could be found as the limit To show that (I − A c ) −1 leaves BC(R, R) invariant we can show that if u ∈ BC(R, R), then for each n ∈ N, S n u (u) ∈ BC(R, R). In turn, this is obvious. It is also obvious that for each n the operator S n u is order-preserving. Therefore, if w ≥ v are two elements in BM (R, R), then This completes the proof of the lemma.
Lemma 2.7. Let K satisfy the Standing Assumption. Then, the operator B c maps B r into BC(R, R) and is Lipschitz continuous.
Proof. Given a fixed w ∈ BC(R, [0, r]); x, x 0 ∈ R, we have By the Assumption ii) in the Standing Assumption the above shows that That means, the function B c (w) is continuous, and B c maps BM (R, [0, r]) into BC(R, R). Next, we show its Lipschitz continuity. By the definition of this operator B c we have Now we are ready to prove the main result of this paper. Theorem 2.8. Let K satisfy the Standing Assumption. Assume further that (2.29) Then, for every there exists a monotone traveling wave solution to Eq. (1.4) with speed c connecting 0 and r.
Remark 2.9. It is well known that when s = η i = 0, i = 1, 2, ...L and K is continuous the existence of monotone traveling waves was proved in [30]. When 1 > s > 0, η i = 0, i = 1, 2, ...L, and K is continuous the existence of traveling waves (that may not be monotone) is shown in [8]. While the appearance of nonzero s is to represent the growth due to old seeds in the seed bank in the models considered in [8,18], the nonzero η i , i = 1, 2, ...L and the discontinuity of the kernel K means the degree of irregularity of the migration of the individuals in the population. This said, the condition (2.29) means if the irregularity of the migration is not too big, monotone traveling waves still exist.
Proof. We will prove this theorem by the Monotone Iteration Method, that is, by constructing a monotone sequence of functions that is convergent to a monotone traveling wave solution. We define a sequence of functions {φ n } ∞ n=1 as follows: Set φ 1 (s) := a(c; s). Since a(c; ·) is non-increasing and bounded it is a measurable and bounded function on R, that is, a(c; ·) ∈ B r . Therefore, we can define (2.31) We will show that this sequence is non-increasing, so convergent. In fact, by definition of the sequence {a n (c; ·)} we have That is, Since A c and B c are order-preserving, by the definition of φ n+1 in (2.31), (2.35) yields that φ n ≥ φ n+1 , n ∈ N.
(2.36) Therefore, for every fixed x ∈ R, the numerical sequence {φ n (x)} is non-increasing and bounded below by zero (as they are all non-negative), so it is convergent to a function, say W (x). By the definitions, The limit function W (x) is a non-increasing function, so it is measurable. Moreover, by the Lebesgue's Dominated Convergence Theorem, On the other hand, by the definition of A c the pointwise limit also implies that in the function space BM (R, [0, r]), it has similar properties as Q as listed in Lemma 2.3. We will show that W is a traveling wave solution to Eq. (1.4). By the general theory in [30] a(c; −∞) = r; a(c; +∞) = 0.
We are going to prove that the boundary conditions for W are satisfied, that is, lim t→−∞ W (t) = r; lim t→+∞ W (t) = 0.
3. Discussion. In this paper we assume that the fecundity function F (x) = xe r0−x , where 0 < r 0 ≤ 1. The condition that 0 < r 0 ≤ 1 is to guarantee the monotonicity of the dynamical system. A large class of functions F could be considered in the same way. For example, any function F that is defined on R and satisfies the following i) g(0) = 0, g(r) = r for a positive constant 0 < r, where g(u) := θF (u) + su, where θ is defined as in (2.4) ; there is no other fixed point of g between 0 and r; ii) F is Lipschitz continuous and increasing on the interval [0, r];