Spreading Speed, Traveling Waves, and Minimal Domain Size in Impulsive Reaction–Diffusion Models

Abstract

How growth, mortality, and dispersal in a species affect the species’ spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction–diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively, they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the population at the end of a pulse as a possibly nonmonotone function of the density of the population at the beginning of the pulse. The dynamics in the dispersal stage is governed by a nonlinear reaction–diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species’ spreading speeds, traveling wave speeds, as well as minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also give an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results.

Appendix

In this section we provide a justification for nonlinearity of g and the proofs for Theorems 2.1, 2.2 and 3.1.

1.1 6.1 Justification of Nonlinearity of g

We consider a population with the following properties: (i) an individual needs a unit of resource to produce offspring; and (ii) each resource unit occupies area b and the mean density of individuals in space is N. Suppose that the population is randomly distributed in space via a Poisson process and individuals are sessile in productive stage. Then the number of individuals in each unit is a Poisson RV with mean Nb. The probability of k individuals in a unit is

$$(Nb)^k e^{-Nb}/k!. $$

Suppose that there is contest competition. If there is more than one individual per unit, then they compete so that only one reproduces. Then the expected number of offspring arising from the unit would be R times the probability that the unit is occupied, where R is the number offspring per adult:

$$R \bigl(1- e^{-bN}\bigr). $$

This is the Skellam function (Skellam 1951), which is qualitatively similar to the Beverton–Holt function (21).

We next assume that there is scramble competition. If two or more individuals chose the same unit, then they would each get a smaller amount of resource, but not enough to reproduce, and so none would reproduce. In this case, the expected number of offspring arising from a unit would be R time the probability that there was one individual occupying the location:

$$R bN e^{-bN}. $$

Choosing r=log(Rb) yields the Ricker function (22).

The first model would apply to animals such as birds which have contest competition for nesting sites, and the second to animals such as salmon, where they spawn in river beds and can spawn on top of a previous site.

1.2 6.2 Proof of Theorem 2.1

Let Q denote the time one solution operator of the reaction–diffusion equation in (1). It is well known that Q is continuous and compact in the topology of uniform convergence on every bounded interval, and Q is monotone in the sense Q[u](x)≥Q[v](x) if u(x)≥v(x)≥0. N _n (x) satisfies the abstract recursion

$$ N_{n+1}(x)=Q\bigl[g (N_n)\bigr](x). $$

(23)

Consider the initial value problem

$$ \everymath{\displaystyle} \begin{array} {@{}l} \frac{\partial p}{\partial t}=d \frac{\partial p}{\partial x^2}+ \alpha p-\gamma p^2, \quad -\infty<x<\infty, \\[4mm] p(x, 0)=\rho N(x). \end{array} $$

(24)

Then v:=p/ρ where 0<ρ≤1 satisfies

$$ \everymath{\displaystyle} \begin{array} {@{}l} \frac{\partial v}{\partial t}=d\frac{\partial v}{\partial x^2}+ \alpha v- \gamma\rho v^2, \quad -\infty<x<\infty, \\[4mm] v(x, 0)=N(x). \end{array} $$

(25)

We introduce the system

$$ \everymath{\displaystyle} \begin{array} {@{}l} \frac{\partial u}{\partial t}=d\frac{\partial u}{\partial x^2}+ \alpha u- \gamma u^2, \quad -\infty<x<\infty, \\[4mm] u(x, 0)= N(x). \end{array} $$

(26)

Since 0<ρ≤1, the comparison theorem shows that the solution v(x,t) of (25) and the solution u(x,t) of (26) satisfy v(x,t)≥u(x,t) for t>0, and particularly,

$$v(x, 1)\geq u(x,1), $$

so that p(x,1)≥ρu(x,1) where p(x,t) is the solution of (24). This shows that Q[ρN](x)≥ρQ[N](x). On the other hand, Hypothesis 2.1 ii implies that for 0<ρ<1, g(ρN)≥ρg(N) for N≥0. It follows that for 0<ρ≤1,

$$ Q\bigl[g(\rho N)\bigr](x)\geq\rho Q\bigl[g(N)\bigr](x). $$

(27)

We use M[N] to denote the linearization of Q[g(N)] about 0. M is the solution operator of the problem

$$ \everymath{\displaystyle} \begin{array} {@{}l} \frac{\partial u}{\partial t}= \frac{\partial u}{\partial x^2} +\alpha u, \quad -\infty<x<\infty, \\[4mm] u(x, 0)=g'(0) N_n(x), \\[2mm] N_{n+1}(x)=u(x, 1). \end{array} $$

(28)

Solving the linear problem (28) explicitly, we obtain that

$$N_{n+1}(x)=M[N_n](x):=\int_{-\infty}^{+\infty}k(x-y)g'(0) e^{\alpha}N_n(y)\,dy $$

where k(x) is the normal distribution given by

$$k(x)=\frac{1}{\sqrt{2\pi d}}e^{-\frac{x^2}{4 d}}. $$

The moment generating function of k(x) is

$$K(x)=\int_{-\infty}^{\infty}e^{\mu x}k(x)\,dx =e^{d\mu^2}, $$

so that

(29)

The condition g(N)≤g′(0)N and the fact that αu−γu ²≤αu, as well as the comparison theorem show that

$$Q\bigl[g( N)\bigr](x)\leq M[N](x). $$

On the other hand, differentiability of g at 0 and Lemma 4.1 in Weinberger et al. (2002) show that there exists a family M ^(κ) of linear order preserving operators with the properties that for every sufficiently large positive integer κ there is a constant ω>0 such that Q[g(v)](x)≥M ^(κ)[v](x) for 0≤v(x)≤ω and that for every μ>0, $B^{(\kappa)}_{\mu}$ defined by B ^(κ) α:=M ^(κ)[e ^−μx α]| _x=0 for all α>0 converge to g′(0)e ^α K(μ) as κ→∞. It follows from Theorems 6.1–6.5 in Weinberger (1982) that system (23) has the spreading speed c ^∗ given by (29) satisfying (9) and (10).

The second part of the theorem follows immediately from Theorem 6.6 in Weinberger (1982). The proof is complete.

1.3 6.3 Proof of Theorem 2.2

A comparison argument that makes use of property (15), similar to the proof of Proposition 3.1 in Li et al. (2009), shows that c ^∗ is the spreading speed of (1). We shall omit the details here.

The nonexistence of traveling wave solutions with speeds c<c ^∗ is similar to the last part of the proof of Theorem 4.1 in Li et al. (2009) and is omitted here.

We now establish the existence of traveling wave solutions with speeds c≥c ^∗. Hypothesis 2.1 ii and the definition of g ⁺ imply that

$$g^+(\rho N)\geq\rho g^+(N) $$

so that (27) with g replaced by g ⁺ holds, i.e.,

$$Q\bigl[g^+ (\rho N)\bigr](x)\geq\rho Q\bigl[g^+(N)\bigr](x). $$

The results about the traveling wave solutions obtained in Sect. 2.2 show that for c≥c ^∗, system (13) has a nonincreasing traveling wave solution N _n (x)=w ⁺(x−nc) with w ⁺(−∞)=β ⁺ and w ⁺(∞)=0. We now choose a positive number 0<ρ<1 so small such that ρw ⁺(x)≤σ ₀ and g(ρw ⁺(x))≤σ ₀. It follows that for N(x)≥ρw ⁺(x):

On the other hand, for 0≤N(x)≤w ⁺(x),

$$Q\bigl[g(N)\bigr](x)\leq Q\bigl[g^+(N)\bigr](x)\leq Q\bigl[g\bigl(w^+\bigr) \bigr](x)=w^+(x-c). $$

We therefore have that the set

$$E_c=\bigl\{ u(x): u(x) \mbox{\ is\ continuous},\ \rho w^+(x)\leq u(x) \leq w^+(x) \bigr\} $$

is an invariant set for the operator T _c [Q[g(⋅)]] with T _c [u](x)=u(x+c). Since g is continuous and Q is compact, the composition operator Q[g(⋅)] is compact. It follows that the image of E _c under T _c [Q[g(⋅)]] is compact in the topology of uniform convergence on every bounded interval. Because the set of bounded vector-valued functions with this topology is a locally convex topological vector space, the existence of a solution w of the equation T _c [Q[g(w)]]=w follows from what Rudin (1991) calls the Schauder–Tychonoff fixed point theorem. Clearly, w(∞)=0. An argument similar to what in the second paragraph on p. 332 in the proof of Theorem 4.1 in Li et al. (2009) shows lim inf _x→∞ w(x)≥β ⁻. The proof is complete.

1.4 6.4 Proof of Theorem 3.1

Consider the eigenvalue problem

$$\everymath{\displaystyle} \begin{array} {@{}l} d\frac{d^2 u}{\partial x^2} +\alpha u= \lambda u, \quad 0<x<\ell, \\[4mm] u(0, t)=u(\ell, t)=0. \end{array} $$

It is easily seen that

$$\lambda_1=\alpha-d\pi^2/\ell^2 $$

is the principal eigenvalue and a corresponding eigenfunction is

$$\phi(x)=\sin\frac{\pi}{\ell}. $$

Let

$$\tilde{N}_n(x)=\kappa\bigl(g'(0) e^{\lambda_1} \bigr)^n \phi(x), \quad n=0, 1, \ldots $$

where κ is a positive constant. $\tilde{N}_{n}(x)$ is a solution of the linear problem

$$ \everymath{\displaystyle} \begin{array} {@{}l} \frac{\partial u}{\partial t}=d \frac{\partial^2 u}{\partial x^2} +\alpha u, \quad 0<x<\ell, \\[4mm] u(0, t)=u(\ell, t)=0, \\[2mm] u(x, 0)=g'(0) \tilde{N}_n(x), \\[2mm] \tilde{N}_{n+1}(x)=u(x, 1). \end{array} $$

(30)

In fact, $u(t,x)=\kappa g'(0) e^{\lambda_{1} t} \phi(x)$ satisfies the linear reaction–diffusion equation and the boundary condition in (30), as well as the initial condition $u(x, 0)=g'(0) \tilde{N}_{0}(x)=\kappa g'(0) \phi(x)$. It follows that

$$u(x,1)=\kappa g'(0) e^{\lambda_1} \phi(x) $$

which is $\tilde{N}_{1}(x)$. Induction shows that $\tilde{N}_{n}(x)$ is a solution of (30).

For any given initial value function u(x,0)=N ₀(x) in (16), we can choose κ sufficiently large such that $N_{0}(x)\leq\tilde{N}_{0}(x)$. Since the reaction term in Eq. (16) is no greater than that in (30) for nonnegative u, the comparison theorem and induction show that the solution N _n (x) of (16) has the property that $N_{n}(x)\leq\tilde{N}_{n}(x)$ for all n≥0. If $g'(0) e^{\lambda_{1}}<1$, then $\lim_{n\rightarrow \infty}\tilde{N}_{n}(x)=0$ for all x, and thus

$$\lim_{n\rightarrow\infty}{N}_n(x)=0. $$

for all x. The proof of the statement (i) is complete.

We now assume $g'(0) e^{\lambda_{1}}>1$ or equivalently $\ell>\pi \sqrt{\frac{d}{\ln g'(0) +\alpha}}$ and prove the statement (ii). We choose $\hat{\lambda}<\lambda_{1}$ and ρ ₁<g′(0) such that $\rho_{1} e^{\hat{\lambda}}>1$. Let $v(x,t)=\epsilon\rho_{1} e^{\hat{\lambda} t}\phi(x)$. It follows from Hypothesis 3.1 that for sufficiently small ϵ>0 and 0<t≤1

On the other hand, for 0<t≤1

for sufficiently small ϵ. This shows that for 0<t≤1, v(x,t) is a lower solution of

$$\everymath{\displaystyle} \begin{array} {@{}l} \frac{\partial u}{\partial t}=d \frac{\partial^2 u}{\partial x^2} +\alpha u-\gamma u^2, \\[4mm] u(0, t)=u(\ell, t)=0. \end{array} $$

We use S to denote the time 1 solution map of

$$\everymath{\displaystyle} \begin{array} {@{}l} \frac{\partial u}{\partial t}=d \frac{\partial u}{\partial x^2} +\alpha u-\gamma u^2 , \\[4mm] u(0, t)=u(\ell, t)=0, \\[2mm] u(x, 0)=u_0(x). \end{array} $$

Then u(x,1)=S[u ₀](x). A comparison argument shows that S is a monotone operator in the sense that S[u ₁](x)≥S[u ₂](x) whenever u ₁(x)≥u ₂(x)≥0. Using S, we find that the solution N _n (x) of (16) satisfies the abstract recursion

$$N_{n+1}(x)=S\bigl[g (N_n)\bigr](x). $$

Let $\underline{N}_{0}(x)=\epsilon\phi(x)$ and $\underline{N}_{n+1}(x)=S[g(\underline{N}_{n})](x)$. The properties of v(x,t) show that for sufficiently small ϵ

$$S\bigl[g(\underline{N}_0)\bigr](x)\geq S[\rho_1 \underline{N}_0](x) \geq v(x, 1)\geq\underline{N}_0(x). $$

Induction shows that $\underline{N}_{n+1}(x)\geq \underline{N}_{n}(x)$ for all n≥0.

On the other hand, for sufficiently small ϵ, the equilibrium value $\beta> \epsilon\phi(x)=\underline{N}_{0}(x)$ is a super solution of (16). It follows that

$$\beta\geq\underline{N}_{n+1}(x)\geq\underline{N}_n(x) $$

for all n≥0. We therefore have that $\underline{N}_{n}(x)$ increases to a limit function $\underline{N}(x)$, which is the minimum positive equilibrium solution of (16). If N ₀(x) in (16) is initially nonnegative and is positive on an open subinterval of (0,ℓ), then the strong maximum principle shows that N ₁(x)>0 for 0<x<ℓ. Choose ϵ sufficiently small so that $\underline{N}_{0}(x)=\epsilon\phi(x)< N_{1}(x)$. Then the comparison theorem shows that $\underline{N}_{n+1}(x)\leq N_{n}(x)$ for all n≥0, and thus $\liminf_{n\rightarrow \infty}\underline{N}_{n}(x)\geq\underline{N}(x)$. The proof is complete.