Monotone traveling waves in a general discrete model for populations with long term memory

. In this paper we consider the existence of monotone traveling waves for a class of general integral difference models for populations that are dependent on the previous state term and also on long term memory. This allows us to consider multiple past states. For this model we will have to deal with the non-compactness of the evolution operator when we prove the existence of a fixed point. This difficulty will be overcome by using the Monotone Iteration Method and Dini’s Theorem to show uniform convergence of an iterative evolution operator to a continuous wave function


Introduction
Many papers use the following integrodifference model to study the dynamics of certain populations (1.1) Here u n (x) is the density of the population at the location x at time n, x ∈ R is a location in the "habitat" of the population and n ∈ Z is the observable time.
In the above mentioned model, given the "fecundity function" f and the diffusion kernel K, one assumes that the dynamics of the system depends only on its status at the last time.A more realistic model should reflect the effect of the history of the system in the past rather than only at the last time.For simplicity, we assume that it depends on the status of the system at time n and n − 1 as other more general settings could be treated in a similar manner.This allows us to consider long term memory of the following evolution operator.
where K i , f i are functions of the same nature as above, though they may be different.
Pioneering works by Weinberger, and contempories studied spreading speed and asymptotic behavior of wave front solutions in population models, see [1,2,8,10,11,[32][33][34]. Furthermore, delay models are often considered in population dynamics, this is due to the fact populations often are reliant on previous states in time, see the manuscripts [3,35].The spreading speed of wavefront solutions have been studied in several types of models.For example, the dynamics in certain competition and cooperation models were studied in [8,11,20,21,34].
In Li, [9] and Thuc, and Nguyen, [7], a version of the following model was studied 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.This is a model for seed banks introduced by MacDonald and Watkinson, [19].The spreading speed and asymptotic behavior for nonlinear integral equations was also studied by Thieme, [26,27].
This paper is interested in dynamics in certain plant populations.In particular, we are interested in monotone wave fronts for Equation (1.2).In section 2, we show the existence of traveling waves for an integrodifferential model of the form Equation (1.2) loses compactness, so standard fixed point theorems do not suffice.To get around this issue we will use Dini's Theorem to show uniform convergence of an evolution operator to a continuous wave front.This is similar to the results found in Thuc and Nguyen, [7].
However, instead of considering the Ricker function, we consider a general Lipchitz function that is bounded on R and is increasing on a certain interval.Furthermore, we strengthen the results by eliminating the continuity condition found the Standing Assumption of the kernel function found in Thuc and Nguyen, [7].Lastly, in section 3 we provide a brief discussion of our results and some interesting questions about extending our idea to more general models, including the addition of discrete terms.

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 R ∋ x → α by this number α for convenience if this does not cause any confusion.
The metric on C M is defined by the sup norm.In BM (R, R) we use the natural order defined as u ≤ v if and only if u(x) ≤ v(x) for all x ∈ R.

Main Results
For model (1.2) the compactness will disappear after a standard conversion of the delayed equation into a non-delayed equation.In fact, by assuming K := K 1 = K 2 and f := f 1 = f 2 , and setting Traveling waves for discrete populations we obtain an equation Next, we denote the projection R 2 ∋ (x, y) T → x ∈ R by P 1 and R 2 ∋ (x, y) T → y ∈ R by P 2 .Then, we obtain the equation we obtain an equation Denote the evolution operator as where F (w n (y)) = f (P 1 w n (y)) + f (P 2 w n (y)) is bounded.Moreover, notice that the projection operator B[w n ](x) = P 2 w n (x) is a constant, so we lose compactness.Thus, we have the following operator We also impose the following conditions on the convolution kernel and fecundity function.
Assume the standing assumptions hold, then the following are true.4. Let w n ∈ BC(R, R) such that w n converges uniformly to some non-negative real constant, w on each bounded subset of R, then A[w n ](x) converges uniformly to A[w](x) for every x ∈ R.
Proof.We only show the increasing portion for i.) since the non-increasing portion can be shown similarly.We note if w(•) is increasing, so is P i (w(•)), i = 1, 2. Furthermore, assume that F (•) is increasing when r ∈ (0, 1), For ii.) A[0] = 0, due to (P 3) of the standing assumption.A[r] = r follows similarly.Take α as some constant, then by (P 4) we have F (α) > α, so the result follows by (P 2).
For iv.) assume w n converges uniformly to a non-negative constant w.By Lebesgue's Dominated Convergence theorem we have Take α > r as some constant, then by (P 4) we have F (α) < α, so v.) follows by (P 2).
It is now possible to define the following operator R c [•] on the space C r for a speed, c as Moreover, define an iterative sequence From [7,32] the sequence {a n (c; •)} is bounded and non-increasing for all s ∈ R. Therefore, we obtain the pointwise limit lim n→∞ a n (c; s) = a(c; s).
The spreading speed is defined as Traveling waves for discrete populations where 0 ≤ a(c; s) ≤ r.We can also define a number c * − as Then, we can define a sequence of functions Defining a non-negative, bounded, real-valued measure m(x, dx) where Now, using the theory in [7,9,32] we can define m(x, dx) = K(x)F ′ (0)dx.
This gives, and Proposition 2.2.Assume the standing assumption holds then the spreading speeds c * , c * − are finite.Proof.The result follows from the definition of Eq(2.5, 2.7) and the standing assumption.■

Existence of Traveling Waves
We are now ready to prove our main result.(2.10) This yields the following operator Proof.We first show that A c maps B r into BC(R, R).To this end, fix u ∈ BC(R, [0, r]), x, x 0 ∈ R. Then we can use (P 2).This means the operator A c is uniformly convergent on R. Thus, for any ε > 0 it is possible to find two constants, T, δ > 0, dependent upon ε such that when This means

3 .
For all v ≤ u, where u, v ∈ B r , then A[v] ≤ A[u].

. 11 ) 2 . 4 .
Lemma Assume the standing assumption holds, then A c : B r → BC(R, R) and A c is Lipchitz continuous.