Spread rates of a juvenile-adult population in constant and temporally variable environments

Abstract

The question of how growth, dispersal, and environmental factors affect the persistence and spread of an invasive species is of great importance in spatial ecology. Motivated by the fact that in a species, different development stages may have different vital rates and dispersal characteristics, we propose and study a reaction-diffusion juvenile-adult model, which is a natural extension of the classical Fisher’s equation. We investigate the spread rates of the population if persistent. By comparing our juvenile-adult model with the physically unstructured Fisher model, we find that Fisher equation can be approximated by our juvenile-adult model in several ways. Accordingly, the spreading speed for Fisher’s model represents a special case of that for the juvenile-adult model. We analyze how the vital rates and different dispersal abilities between juveniles and adults influence the spreading spread of the structured population, the results indicate that the juvenile-adult model provides more insights into population spread than Fisher equation. We then study a reaction-diffusion juvenile-adult model with temporally periodic coefficients. We develop a novel numerical method to calculate the spreading speed under temporal variability. Finally, we utilize the time-periodic model to understand the spatial spread of a population with separate breeding and non-breeding seasons. In particular, we scrutinize how the seasonal variation in vital rates and dispersal rates, and the duration of the breeding season affect the spreading speed of the population. The theory developed here can provide effective strategies to control the spread of invasive species.

Appendices

Appendix A

Comparison between the juvenile-adult model and the Fisher’s model

We shall show that the juvenile-adult model (2) reduces to Fisher’s equation in two special scenarios. Accordingly, the spreading speed of the juvenile-adult population is an approximation to that of single-compartment population.

The maturation rate $\textit{\textbf{g}}$ is very large

We show that the limiting system of (2) as $g\rightarrow \infty$ (i.e., individuals can reproduce immediately after being born) is Fisher’s equation. Summing the first and the second equation of (2), we obtain

$$\begin{aligned} \frac{\partial (J+A)}{\partial t}= & {} D_1\frac{\partial ^2 J}{\partial x^2}+D_2\frac{\partial ^2 A}{\partial x^2} \\&-m_1J+bA-m_2A-\alpha _1 J^2-\alpha _2 A^2. \end{aligned}$$

(32)

Since $g\rightarrow \infty$, the first equation of (2) yields $J\rightarrow 0$, accordingly, system (A1) becomes

$$\begin{aligned} \frac{\partial A}{\partial t}=D_2\frac{\partial ^2 A}{\partial x^2}+(b-m_2)A-\alpha _2 A^2. \end{aligned}$$

(33)

That is, as $g\rightarrow \infty$, the juvenile-adult model reduces to an unstructured model because all newborns immediately become adults. Clearly, (A2) is Fisher’s equation, which yields an asymptotic spreading speed $2\sqrt{(b-m_2)D_2}$. Note that as $g\rightarrow \infty$, the persistence condition (4) becomes $b>m_2$ as required.

Both stages have the same life characteristics

Due to ‘linear determinacy’ for spread in cooperative models (Weinberger et al. (2002)), which equates the spreading speed in a nonlinear system (2) with the minimum traveling wave speed of the linearized (at zero) system if no Allee effect is involved, we focus on the linearized system

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial J}{\partial t}=D_1\frac{\partial ^2 J}{\partial x^2}+bA-gJ-m_1J,\\ \frac{\partial A}{\partial t}=D_2\frac{\partial ^2 A}{\partial x^2}+gJ-m_2A. \end{array}\right. } \end{aligned}$$

(34)

We now show that the linearized juvenile-adult model (A3) reduced a linearized Fisher’s equation when the juvenile and the adult have the same mortality rate and the same dispersal ability. Letting $m_1=m_2=:m$ and $D_1=D_2{:=}D$, we add the two equations of (A3) together to get

$$\begin{aligned} \frac{\partial (J+A)}{\partial t}=D\frac{\partial ^2 (J+A)}{\partial x^2}+bA-m(J+A). \end{aligned}$$

(35)

We introduce a density-dependent reproduction rate $\hat{b}(J,A)=\frac{bA}{J+A}$. Then (A4) can be written as

$$\begin{aligned} \frac{\partial (J+A)}{\partial t}= & {} D\frac{\partial ^2 (J+A)}{\partial x^2} \\&+\hat{b}(J,A)(J+A)-m(J+A). \end{aligned}$$

(36)

We then approximate $\hat{b}(J,A)$ by a constant reproduction rate, which can be done by considering a stable ratio of J to A. The spatially homogeneous system of (A3) with $m_1=m_2=m$ and $D_1=D_2{:=}D$ is given by

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d J}{d t}=bA-gJ-mJ,\\ \frac{d A}{d t}=gJ-mA. \end{array}\right. } \end{aligned}$$

(37)

Straightforward calculations give

$$\begin{aligned} \frac{d}{dt}\left( \frac{J}{A}\right) =\frac{1}{A}\frac{dJ}{dt}-\frac{J}{A^2}\frac{dA}{dt} =b-g\left( \frac{J}{A}\right) -g\left( \frac{J}{A}\right) ^2. \end{aligned}$$

(38)

Setting $\frac{d}{dt}\left( \frac{J}{A}\right) =0$, we find a stable ratio of J to A given by

$$\begin{aligned} \sqrt{\frac{1}{4}+\frac{b}{g}}-\frac{1}{2}{:=}\frac{\bar{J}}{\bar{A}}, \end{aligned}$$

which is the same as the asymptotically stable ratio we found in section 2.2 (see (22)). Thus, we approximate $\hat{b}(J,A)$ by

$$\begin{aligned} \frac{b\bar{A}}{\bar{J}+\bar{A}}=\frac{b}{\bar{J}/\bar{A}+1} =\frac{2b}{\sqrt{1+\frac{4b}{g}}+1}{:=}\bar{b}. \end{aligned}$$

Therefore, (36) can be approximated by

$$\begin{aligned} \frac{\partial (J+A)}{\partial t}=D\frac{\partial ^2 (J+A)}{\partial x^2}+\bar{b}(J+A)-m(J+A), \end{aligned}$$

(39)

which is a linearized Fisher’s equation with a spreading speed $2\sqrt{(\bar{b}-m)D}$. Notice that when $m_1=m_2=m$, the persistence condition (4) becomes $bg>(g+m)m$, which is exactly equivalent to $\bar{b}>m$ as required.

Appendix B

A numerical method for computing $\varvec{\rho (\mu )}$

We divide the interval [0, T] into n equal subintervals $[t_i, t_{i+1}]$, $i=1,2, \cdots , n$, where $0=t_1<t_2<\cdots <t_{n+1}=T$. Denote the length of subintervals by $\Delta t=T/n$. We then discretize system (27) by the system of difference equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\mathcal {J}(t_{i+1})-\mathcal {J}(t_i)}{\Delta t}=[D_1(t_i)\mu ^2-g(t_i)+m_1(t_i))]\mathcal {J}(t_i)+b(t_i) \mathcal {A}(t_i),\\ \frac{\mathcal {A}(t_{i+1})-\mathcal {A}(t_i)}{\Delta t}=g(t_i)\mathcal {J}(t_i)+[D_2(t_i)\mu ^2-m_2(t_i)]\mathcal {A}(t_i). \end{array}\right. } \end{aligned}$$

(40)

Introducing the matrix

$$\begin{aligned} \begin{pmatrix} \Delta t[D_1(t_i)\mu ^2-g(t_i)+m_1(t_i))]+1 &{} \Delta t b(t_i) \\ \Delta tg(t_i) &{} \Delta t[D_2(t_i)\mu ^2-m_2(t_i)]+1 \\ \end{pmatrix}, \ \ i=1,2,\cdots , n, \end{aligned}$$

we rewrite (B1) as

$$\begin{aligned} \begin{pmatrix} \mathcal {J}(t_{i+1}) \\ \mathcal {A}(t_{i+1}) \\ \end{pmatrix}=M_i \begin{pmatrix} \mathcal {J}(t_{i}) \\ \mathcal {A}(t_{i}) \\ \end{pmatrix}, i=1,2,\cdots , n. \end{aligned}$$

Thus, if $\Delta t$ is sufficiently small, we have the following approximation:

$$\begin{aligned} \begin{pmatrix} \mathcal {J}(T) \\ \mathcal {A}(T) \\ \end{pmatrix}\approx M_n\cdots M_2 M_1 \begin{pmatrix} \mathcal {J}(0) \\ \mathcal {A}(0) \\ \end{pmatrix}. \end{aligned}$$

In terms of the definition of the Poincaré map $Q_\mu ^T$ of (28), we can approximate $Q_\mu ^T$ by the matrix operator $M_n\cdots M_2 M_1{:=}\mathcal {M}_n$ by choosing a sufficiently large n. Thus, $\rho (\mu )$, the spectral radius of $Q_\mu ^T$, can be approximated by the spectral radius of the matrix $\mathcal {M}_n$, for very large n.

We observe that if the functions b(t) and g(t) are positive, then when $\Delta t$ is sufficiently small, all entries of the matrix $M_i$ are positive, so all entries of the matrix $\mathcal {M}_i$ are positive as well. By the famous Perron-Frobenius Theorem, we have the following results: (1) $\mathcal {M}_n$ has a positive real eigenvalue $\hat{\rho }(\mu )_n$, called the dominant eigenvalue, such that any other eigenvalue (possibly, complex) is strictly smaller than $\hat{\rho }(\mu )_n$, in absolute value. Therefore, the spectral radius of matrix $\mathcal {M}_n$ is equal to $\hat{\rho }(\mu )_n$. (2) There exists an eigenvector $(\xi _{n,1}, \xi _{n,2})$ of $\mathcal {M}_n$ associated with eigenvalue $\hat{\rho }(\mu )_n$ such that both components of this eigenvector are positive. In other words, the eigenvector $(\xi _1, \xi _2)$ associated with $\rho (\mu )$ can be approximated by $(\xi _{n,1}, \xi _{n,2})$ for very large n.