How Phenological Variation Affects Species Spreading Speeds

Abstract

In this paper, we develop a phenologically explicit reaction–diffusion model to analyze the spatial spread of a univoltine insect species. Our model assumes four explicit life stages: adult, two larval, and pupa, with a fourth, implicit, egg stage modeled as a time delay between oviposition and emergence as a larva. As such, our model is broadly applicable to holometabolous insects. To account for phenology (seasonal biological timing), we introduce four time-dependent phenological functions describing adult emergence, oviposition and larval conversion, respectively. Emergence is defined as the per-capita probability of an adult emerging from the pupal stage at a particular time. Oviposition is defined as the per-capita rate of adult egg deposition at a particular time. Two functions deal with the larva stage 1 to larva stage 2, and larva stage 2 to pupa conversion as per-capita rate of conversion at a particular time. This very general formulation allows us to accommodate a wide variety of alternative insect phenologies and lifestyles. We provide the moment-generating function for the general linearized system in terms of phenological functions and model parameters. We prove that the spreading speed of the linearized system is the same as that for nonlinear system. We then find explicit solutions for the spreading speed of the insect population for the limiting cases where (1) emergence and oviposition are impulsive (i.e., take place over an extremely narrow time window), larval conversion occurs at a constant rate, and larvae are immobile, (2) emergence and oviposition are impulsive (i.e., take place over an extremely narrow time window), larval conversion occurs at a constant rate starting at a delayed time from egg hatch, and larvae are immobile, and (3) emergence, oviposition, and larval conversion are impulsive. To consider other biological scenarios, including cases with emergence and oviposition windows of finite width as well as mobile larvae, we use numerical simulations. Our results provide a framework for understanding how phenology can interact with spatial spread to facilitate (or hinder) species expansion. This is an important area of research within the context of global change, which brings both new invasive species and range shifts for native species, all the while causing perturbations to species phenology that may impact the abilities of native and invasive populations to spread.

Appendices

Appendix A: Proof of Theorem 3.1

We demonstrate that model (1) satisfies Hypotheses 2.1 and the conditions given in Theorem 3.1 in Weinberger et al. (2002), so that the spreading speed of the nonlinear model (1) is the same as that of its linearization.

We show that P(T, x) in (1) is mathematically well defined. There are two cases: case (i) where $g(t),\,r(t),\,m_i(t)$ are all bounded and piecewise continuous functions, and case (ii) where one of $g(t),\,r(t),\,m_i(t)$ is a Dirac delta distribution. We first consider case (i) and assume that all phenological functions are bounded and continuous functions. If $P_n(x)$ is continuous and bounded, Theorem 2.2 in Pao (1992) implies that the first equation in (1) has a unique solution A(t, x) which is continuous and bounded in t and x. The same theorem then implies that the second, third, and fourth equation of (1) have unique solutions $L_1(t,x)$, $L_2(t,x)$ and P(t, x) that are continuous and bounded in t and x, respectively, and thus $P_{n+1}( x)=P(T, x)$ is bounded and continuous in x. Since $P_0(x)$ is continuous and bounded, induction shows the continuity and boundedness of $P_n(x)$ for all $n\ge 1$. If g(t) is piecewise continuous and discontinuous at $0<s_1<s_2<\cdot \cdot \cdot<s_{\ell }<T$, the first equation of (1) shows the existence, uniqueness, and boundedness of A(t, x) for $0<t\le s_1$. One can then consider the first equation of (1) for $s_1<t\le s_2$ with the initial value $A(s_1, x)$ at $s_1$ to establish the existence, uniqueness, and boundedness of A(t, x) for $s_1<t\le s_2$. Induction shows the existence, uniqueness, and boundedness of A(t, x) for $0<t\le T$. The same method can be used to determine the existence, uniqueness, and boundedness of $L_i(t, x)$ and P(t, x) for $\tau <t\le T$ if r(t) or $m_i(t)$ is piecewise continuous.

We now consider case (ii). If one of $g(t),\,r(t),\,m_i(t)$ is a Dirac delta function, the corresponding equation is converted into an autonomous equation with an appropriate initial condition.

If $g(t)=\delta (t-t_{emg})$, then the equation for A(t, x) in (1) is equivalent to the following classical system:

$$\begin{aligned} {\left\{ \begin{array}{ll} A(t,x)=0,&{}0\le t<t_{emg},\\ \frac{\partial }{\partial t}A(t,x)=d_a\frac{\partial ^2}{\partial x^2}A-\nu _aA-\beta _aA^2,&{}t_{emg}<t\le T,\\ A(t_{emg},x)=\alpha _1P_n(x).\\ \end{array}\right. } \end{aligned}$$

Theorem 2.2 in Pao (1992) again works to show the existence and uniqueness of a continuous and bounded solution for (1).

Similarly, if $r(t)=\alpha _2\delta (t-t_o)$ ($t_o>t_{emg}$ if $g(t)=\delta (t-t_{emg})$), then the equation for $L_1(t,x)$ in (1) is equivalent to the following classical system:

$$\begin{aligned} {\left\{ \begin{array}{ll} L_1(t,x)=0,&{}\tau \le t<t_o+\tau ,\\ \frac{\partial }{\partial t}L_1(t,x)=d_{l_1}\frac{\partial ^2}{\partial x^2}L_1-\nu _{l_1}L_1-\beta _{l_1}L_1^2,&{}t_o+\tau <t\le T,\\ L_1(t_o+\tau ,x)=\alpha _2A(t_o,x).\\ \end{array}\right. } \end{aligned}$$

For $m_1(t)=\gamma _1\,\delta (t-t_{l_1})$, pupation occurs impulsively at time $t_{l_1}$ and therefore $L_1(t,x)$ is not continuous with respect to time at $t=t_{l_1}$. In this case, the $L_1$ and $L_2$ equations in (1) may be viewed as

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial }{\partial t}L_1(t,x)=d_{l_1}\frac{\partial ^2}{\partial x^2}L_1-\nu _{l_1}L_1-\beta _{l_1}L_1^2+r(t-\tau )A(t-\tau ,x), &{}\tau<t<t_{l_1}, \\ L_1(\tau ,x)=0, &{}\\ \frac{\partial }{\partial t}L_1(t,x)=d_{l_1}\frac{\partial ^2}{\partial x^2}L_1-\nu _{l_1}L_1-\beta _{l_1}L_1^2+r(t-\tau )A(t-\tau ,x), &{}t_{l_1}<t\le T, \\ L_1(t_{l_1}^+,x)=e^{-\gamma _1}L_1(t_{l_1}^-,x), \end{array}\right. } \\ {\left\{ \begin{array}{ll} L_2(\tau ,x)=0, &{}\tau<t<t_{l_1}\\ \frac{\partial }{\partial t}L_2(t,x)=d_{l_2}\frac{\partial ^2}{\partial x^2}L_2-\big (\nu _{l_2}+m_2(t)\big )L_2-\beta _{l_2}L_2^2, &{}t_{l_1}<t\le T, \\ L_2(t_{l_2}^+,x)=(1-e^{-\gamma _1})L_1(t_{l_1}^-,x), \end{array}\right. } \end{aligned} \end{aligned}$$

(7)

To see $L_1(t_{l_1}^+,x)=e^{-\gamma _1}L_1(t_{l_1}^-,x)$, in (1), we replace $m_1(t)$ by its approximation $m_\lambda (t)=\frac{m_0}{\lambda }\phi \left( \frac{t-t_3}{\lambda }\right)$ where $\phi$ is a nonnegative bounded continuous function with support $[-1,1]$ and $\int ^{1}_{-1}\phi (s)\mathrm {d}s=1$, integrate the $L_1$ equation from $t^-$ to $t^+$ with $t^-<t_{l_1}<t^+$, and use the boundedness and second order differentiability of the solution and take the limit as $\lambda {\rightarrow }0^+$, $t^-{\rightarrow }t^-_3$ and $t^+{\rightarrow }t^+_3$.

On the other hand, since all stage 1 larva that defect at $t_{l_1}$ are assumed to convert to stage 2 larva, $L_2(t_{l_1}^+,x)=(1-e^{-\gamma _1})L_1(t_{l_1}^-,x)$.

For the case that $m_2(t)=\gamma _2\,\delta (t-t_{l_2})$, the result is analogous to that for $m_1(t)=\gamma _1\,\delta (t-t_{l_1})$, with $P(t_{l_2}^+,x)=\big (1-e^{-\gamma _2}\big )\,L_2(t_{l_2}^-,x)$.

We have justified that the solution to model (1) maps the pupa density distribution at the end of the production season from year n, $P_{n}(x)$, to the pupa density distribution at the end of the production season from year $n+1$, $P_{n+1}(x)$. Mathematically model (1) can be rewritten as an abstract discrete equation in the form of $P_{n+1}(x)=Q[P_{n}](x)$ with $Q[P_{n}](x)$ determined by P(T, x).

Clearly, $P_0(x)\equiv 0$ implies $P_n(x)\equiv 0$ for all n. That is $Q[0]=0$. The comparison principals for scalar parabolic equations and ordinary differential equations show that $Q[p](x)\ge Q[q](x)$ if $p(x)\ge q(x)\ge 0$. That is, Q is a monotone operator. One can easily see that for the ODE model corresponding to (1), a sufficiently large number M is an upper solution. This M is also an upper solution for the PDE model (1), i.e., $Q[M]\le M$. It follows from elementary properties of parabolic systems that Q is translation and reflection invariant, and Q is continuous in the topology of uniform convergence on bounded sets. Hypotheses 2.1 (i)–(iv) given in Weinberger et al. (2002) are satisfied by Q if Q has a positive equilibrium.

The linearized system of model (1) is given by

$$\begin{aligned} \frac{\partial }{\partial t} A&=d_a\frac{\partial ^2}{\partial x^2}A-\nu _aA+\alpha _1g(t)P_{n}(x), \&0<t\le T,\nonumber \\ \frac{\partial }{\partial t} L_1&=d_{l_1}\frac{\partial ^2}{\partial x^2} L_1-(\nu _{l_1}+m_1(t))L_1+r(t-\tau )A(t-\tau ,x), \&\tau<t\le T,\nonumber \\ \frac{\partial }{\partial t} L_2&=d_{l_2}\frac{\partial ^2}{\partial x^2} L_2-(\nu _{l_2}+m_2(t))L_2+m_1(t)L_1, \&\tau<t\le T,\nonumber \\ \frac{\partial }{\partial t}P&=m_2(t)L_2-\nu _pP, \&\tau <t\le T,\nonumber \\&\quad A(0,x)=0, \ \ L_1(\tau ,x)=L_2(\tau ,x)=P(\tau ,x)=0,\nonumber \\&\quad P_{n+1}(x)=P(T,x). \end{aligned}$$

(8)

To find a solution to model (8) subject to same initial conditions as model (1), we will solve for A(t, x) and then sequentially work our way to P(t, x). An explanation of how to solve this equation using integral methods can be found in Polyanin (2002). From the first equation of model (8) with initial pupal population $P_n(x)$, we find

$$\begin{aligned} A(t,x)=\alpha _1\int _{0}^{t}\int _{-\infty }^{\infty }\!\frac{1}{2\sqrt{\pi d_a(t-s)}}\exp \left( -\nu _a(t-s)-\frac{(x-\zeta )^2}{4d_a(t-s)}\right) g(s)P_n(\zeta )\,\mathrm {d}\zeta \,\mathrm {d}s.\nonumber \\ \end{aligned}$$

(9)

Substituting this result into the second equation of (8) gives the first stage larval population for the system:

$$\begin{aligned} L_1(t,x)&=\int _{\tau }^{t}\int _{-\infty }^{\infty }\!\frac{1}{2\sqrt{\pi d_{l_1}(t-s)}}\exp \left( -\frac{(x-\zeta )^2}{4d_{l_1}(t-s)}\right) \nonumber \\&\quad \exp \bigg (-\nu _{l_1}(t-s)-M_1(t)+M_1(s)\bigg )\nonumber \\&\quad r(s-\tau )A(s-\tau ,\zeta )\,\mathrm {d}\zeta \,\mathrm {d}s, \end{aligned}$$

(10)

where $M_1(t)$ is an anti-derivative of $m_1(t)$. The second stage larva density is given by

$$\begin{aligned} L_2(t,x)&=\int _{\tau }^{t}\int _{-\infty }^{\infty }\!\frac{m_1(s)}{2\sqrt{\pi d_{l_2}(t-s)}}\exp \left( -\frac{(x-\zeta )^2}{4d_{l_2}(t-s)}\right) \nonumber \\&\quad \exp \bigg (-\nu _{l_2}(t-s)-M_2(t)+M_2(s)\bigg )\nonumber \\&\quad L_1(s,\zeta )\,\mathrm {d}\zeta \,\mathrm {d}s, \end{aligned}$$

(11)

where $M_2(t)$ is an anti-derivative of $m_2(t)$. Finally, the pupal density at time T is given by

$$\begin{aligned} P(T, x)=\int _{\tau }^{T}\!m_2(s)e^{-\nu _p(T-s)}L_2(s,x)\,\mathrm {d}s. \end{aligned}$$

(12)

Equation (12) describes the number of larvae that successfully mature into pupae by the end of the year. Because it is also proportional to the emerging adult population in the following year, Eqs. (9)–(12) completely describe the species life cycle. Consequently, these equations can be combined to give the following year-to-year linear mapping

$$\begin{aligned} P_{n+1}(x)&=\,\alpha _1\int ^{T}_{\tau }\!\mathrm {d}s_4\int ^{s_4}_{\tau }\!\mathrm {d}s_3\int ^{s_3}_{\tau }\!\mathrm {d}s_2\int ^{s_2-\tau }_{0}\!\mathrm {d}s_1\nonumber \\&\quad \times \int ^{\infty }_{-\infty }\!\mathrm {d}\zeta _3\int ^{\infty }_{-\infty }\!\mathrm {d}\zeta _2\int ^{\infty }_{-\infty }\!\mathrm {d}\zeta _1\;m_2(s_4)\,m_1(s_3)\,\exp (-\nu _p(s_4-s_3))\nonumber \\&\quad \times \frac{1}{\sqrt{4\pi \,d_{l_2}(s_4-s_3)\,}}\,\frac{1}{\sqrt{4\pi \,d_{l_1}(s_3-s_2)\,}}\,\frac{1}{\sqrt{4\pi \,d_a(s_2-\tau -s_1)\,}}\nonumber \\&\quad \times \exp \left( -\nu _{l_2}(s_4-s_3)-M_2(s_4)+M_2(s_3)-\frac{(x-\zeta _3)^2}{4d_{l_2}(s_4-s_3)}\right) \nonumber \\&\quad \times \exp \left( -\nu _{l_1}(s_3-s_2)-M_1(s_3)+M_1(s_2)-\frac{(\zeta _3-\zeta _2)^2}{4d_{l_1}(s_3-s_2)}\right) \nonumber \\&\quad \times \exp \left( -\nu _a(s_2-\tau -s_1)-\frac{(\zeta _2-\zeta _1)^2}{4d_a(s_2-\tau -s_1)}\right) r(s_2-\tau )g(s_1)P_{n}(\zeta _1). \end{aligned}$$

(13)

The right-hand side of this recursion defines an linear operator R. That is, (13) can be written as $P_{n+1}(x)=R[P_n](x)$. As indicated in Weinberger et al. (2002), the moment-generating function for R is then given by:

$$\begin{aligned} \Lambda (\mu )&=R[e^{-\mu x}](0)\nonumber \\&=\int ^T_\tau \,\mathrm {d}s_4\int ^{s_4}_\tau \mathrm {d}s_3\int ^{s_3}_\tau \mathrm {d}s_2\int ^{s_2-\tau }_0\mathrm {d}s_1\text {}\alpha _1\,m_2(s_4)\,m_1(s_3)\,r(s_2-\tau )\,g(s_1)\nonumber \\&\quad \exp \Big (\mu ^2\big (d_{l_2}(s_4-s_3)+d_{l_1}(s_3-s_2)+d_a(s_2-\tau -s_1)\big )\Big )\nonumber \\&\quad \exp \Big (-\nu _p(T-s_4)-\nu _{l_2}(s_4-s_3)-\nu _{l_1}(s_3-s_2)-\nu _a(s_2-\tau -s_1)\Big )\nonumber \\&\quad \exp \Big (-M_2(s_4)+M_2(s_3)-M_1(s_3)+M_1(s_2)\Big ) \end{aligned}$$

(14)

The zero solution for R is unstable if $\Lambda (0)>1$.

Assume that $\Lambda (0)>1$. For any positive integer $\kappa >1$, we define $R^{(\kappa )}[P_n](x)$ to be the solution operator of the linear system (8) with the death rate $\nu _i$ replaced by $\nu ^{(\kappa )}_i=\nu _i+\kappa ^{-1}$. Let $\Lambda ^{(\kappa )}(\mu )=R^{(\kappa )}[e^{-\mu x}](0)$. It is easily seen that $\Lambda ^{(\kappa )}(\mu )$ is given by $\Lambda (\mu )$ with $\nu _i$ replaced by $\nu ^{(\kappa )}_i$ for $i\in \{a,\,l_1,\,l_2,\,p\}$. Clearly, the limit of $\Lambda ^{(\kappa )}(\mu )$ as $\kappa$ approaches $\infty$ is $\Lambda (\mu )$. For any $\kappa >1$, there is $\delta _{\kappa }>0$ such that for any $0<\alpha \le \delta _{\kappa }$ such that

$$\begin{aligned}-\nu ^{(\kappa )}_aA \le -\nu _aA-\beta _aA^2, \ \ -\nu ^{(\kappa )}_{l_i}L_i \le -\nu _{l_i}L_i-\beta _{l_i}L_i^2 , \ \ -\nu ^{(\kappa )}_p\le -\nu _pP.\end{aligned}$$

We now observe that $R^{(\kappa )}[\delta _{\kappa }]=\Lambda ^{(\kappa )}(0)\delta _{\kappa }$. Since $\Lambda (0)>1$, $\Lambda ^{(\kappa )}(0)>1$ for large $\kappa$. Choose $\omega =\delta /\Lambda ^{(\kappa )}(0)$. Then, a standard comparison theorem shows that for $0\le v(x) \le \omega$, $Q[v](x)\ge R^{(\kappa )}[v](x)$. We have verified Hypothesis 2.1.vi in Weinberger et al. (2002). Hypothesis 2.1.v in the paper is automatically satisfied. On the other hand, $Q[v](x)\ge R^{(\kappa )}[v](x)$ for $0\le v(x) \le \omega$ implies that for small positive $\alpha$, $Q[\alpha ]>\alpha$. This together with $Q[M]\le M$ shows that Q has a positive equilibrium $P^*$ satisfying $Q[P^*]=P^*$. We have shown that Hypotheses 2.1 in Weinberger et al. (2002) hold if $\Lambda (0)>1$.

In model (1), the nonlinear quadratic terms are all non-positive. This implies that Q is dominated by R, and thus for the scalar operator Q the conditions in Theorem 3.1 in Weinberger et al. (2002) are automatically satisfied. It follows from this theorem that the spreading speed of Q is the same as that of R, which is given by

$$\begin{aligned} c^{*}:= & {} \inf _{\mu >0} \frac{\ln [\Lambda (\mu )]}{\mu } \end{aligned}$$

(15)

if $\Lambda (0)>1$. On the other hand, since at least one diffusion coefficient is positive, the basic properties of reaction–diffusion equations show that the operator P is compact in the sense that every sequence $v_n(x)$ with $0\le v_n(x)\le P^*$ has a subsequence $v_{n_{\ell }}(x)$ such that $P[v_{n_{\ell }} ](x)$ converges uniformly on every bounded set. It follows from Theorem 3.1 in Li et al. (2005) that $c^*$ is the slowest speed of a class of traveling wave solutions connecting 0 with $P^*$.

Appendix B: Spreading Speed for Impulsive Emergence, Oviposition with no Larval Dispersal, and Constant Pupation Rate

We define $g,\,r,\,m_1,\,m_2$ as

$$\begin{aligned}g(t)=\delta (t),\ r(t)=\alpha _2\,e^{-\nu _e\tau }\,\delta (t-t_2),\ m_1(t)\equiv m_1,\ m_2(t)\equiv m_2\end{aligned}$$

and set $d_{l_1}=d_{l_2}=0$. Applying this to Eq. (4), we arrive at the following integral for the moment-generating function,

$$\begin{aligned}&\Lambda (\mu )=\int ^{T}_{\tau }\mathrm {d}s_4\int ^{s_4}_\tau \mathrm {d}s_3\int ^{s_3}_\tau \mathrm {d}s_2\int ^{s_2-\tau }_0\mathrm {d}s_1\;\alpha _1\alpha _2\,m_1m_2\,e^{-\nu _e\tau }\\&\delta (s_2-t_o-\tau )\,\delta (s_1)\,\exp \!\bigg (\mu ^2d_a(s_2-\tau -s_1)-m_2(s_4-s_3)-m_1(s_3-s_2)\bigg )\\&\exp \!\bigg (-\nu _p(T-s_4)-\nu _{l_2}(s_4-s_3)-\nu _{l_1}(s_3-s_2)-\nu _a(s_2-\tau -s_1)\bigg ). \end{aligned}$$

The $\mathrm {d}s_1$ and $\mathrm {d}{s_2}$ integrals are straightforward with the non-delta portion of the integrand being evaluated at $s_1=0$ and $s_2=t_o+\tau$, respectively. The lower bounds of the $\mathrm {d}s_3$ and $\mathrm {d}s_4$ integral must be replaced with $t_o+\tau$, and then these integrals can be carried out in a straightforward manner. After some simplification, it is found that

$$\begin{aligned} \Lambda (\mu )=&\,\alpha _1\alpha _2\,m_1m_2\,\exp \!\big (\mu ^2d_at_o\big )\,\exp \!\big (-\nu _e\tau -\nu _at_o\big )\\&\quad \times \left( \frac{e^{-(m_1+\nu _{l_1})(T-t_o-\tau )}-e^{-\nu _p(T-t_o-\tau )}}{(m_2-m_1+\nu _{l_2}-\nu _{l_1})(\nu _p-m_1-\nu _{l_1})}\right. \\&\left. \quad -\frac{e^{-(m_2+\nu _{l_2})(T-t_o-\tau )}-e^{-\nu _p(T-t_0-\tau )}}{(m_2-m_1+\nu _{l_2}-\nu _{l_1})(\nu _p-m_2-\nu _{l_2})}\right) . \end{aligned}$$

Invoking the non-extinction condition, $\Lambda (0)>1$, we find the parameters must satisfy

$$\begin{aligned} \alpha _1\alpha _2>\frac{e^{\nu _e\tau }\,e^{\nu _at_o}}{m_1m_2\,\Bigg (\frac{e^{-(m_1+\nu _{l_1})(T-t_o-\tau )}-e^{-\nu _p(T-t_o-\tau )}}{(m_2-m_1+\nu _{l_2}-\nu _{l_1})(\nu _p-m_1-\nu _{l_1})}-\frac{e^{-(m_2+\nu _{l_2})(T-t_o-\tau )}-e^{-\nu _p(T-t_0-\tau )}}{(m_2-m_1+\nu _{l_2}-\nu _{l_1})(\nu _p-m_2-\nu _{l_2})}\Bigg )}. \end{aligned}$$

To find $c^*$, we must find $\inf _{\mu >0}\,\frac{\ln (\Lambda (\mu ))}{\mu }$. Taking the natural log of $\Lambda (\mu )$, we find

$$\begin{aligned} \ln (\Lambda (\mu ))=A+B\mu ^2, \end{aligned}$$

where

$$\begin{aligned} A&:=\ln \big (\alpha _1\alpha _2\,m_1m_2\big )-\nu _e\tau -\nu _at_o\\&\quad \ln \!\left( \frac{e^{-(m_1+\nu _{l_1})(T-t_o-\tau )}-e^{-\nu _p(T-t_o-\tau )}}{(m_2-m_1+\nu _{l_2}-\nu _{l_1})(\nu _p-m_1-\nu _{l_1})}\right. \\&\quad \left. -\frac{e^{-(m_2+\nu _{l_2})(T-t_o-\tau )}-e^{-\nu _p(T-t_0-\tau )}}{(m_2-m_1+\nu _{l_2}-\nu _{l_1})(\nu _p-m_2-\nu _{l_2})}\right) ,\\ B&:=d_at_o. \end{aligned}$$

From elementary calculus, we find $c^*=\inf _{\mu >0}\,\frac{A+B\mu ^2}{\mu }=2\sqrt{AB}$, and thus

$$\begin{aligned} c^*&=2\,\big (d_a\,t_o\,\big )^\frac{1}{2}\\&\quad \times \Big (\ln \!\Big (\alpha _1\,\alpha _2\,m_1\,m_2\Big )-\nu _e\tau -\nu _at_o\\&\quad +\ln \!\left( \frac{e^{-(m_1+\nu _{l_1})(T-t_o-\tau )}-e^{-\nu _p(T-t_o-\tau )}}{(m_2-m_1+\nu _{l_2}-\nu _{l_1})(\nu _p-m_1-\nu _{l_1})}\right. \\&\left. \quad -\frac{e^{-(m_2+\nu _{l_2})(T-t_o-\tau )}-e^{-\nu _p(T-t_0-\tau )}}{(m_2-m_1+\nu _{l_2}-\nu _{l_1})(\nu _p-m_2-\nu _{l_2})}\right) \Big )^{\frac{1}{2}}. \end{aligned}$$

Appendix C: Spreading Speed for Impulsive Emergence, Oviposition, and Larval Conversion

To evaluate the moment generator, we will need to employ the following lemma:

Lemma 4.1

Suppose $f:{\mathbb {R}}\mapsto {\mathbb {R}}$ is continuous at $x=0$, and $g(x):{\mathbb {R}}\mapsto {\mathbb {R}}$ is continuous and nonnegative on [0, 1] then:

$$\begin{aligned}&\int _{-\infty }^{\infty }g\left( \int _{-\infty }^{x}\delta (s)\mathrm {d}s\right) \,f(x)\,\delta (x)\,\mathrm {d}x=f(0)\,\int _{0}^{1}g(x)\mathrm {d}x. \end{aligned}$$

Proof

Let $\phi (x)$ be any function with the following properties: $\phi (x)\in \mathbf{C }({\mathbb {R}})$, $\phi (x)$ is nonnegative, $\phi (x)=0$ if $|x|\ge 1$, $\int _{-\infty }^{\infty }\!\phi (x)\,\mathrm {d}x=1.$

We define $\phi _{\lambda }(x)=\frac{1}{\lambda }\,\phi \left( \frac{x}{\lambda }\right)$ for $\lambda >0$. We note the support of $\phi _{\lambda }(x)$ is a subset of $[-\lambda ,\lambda ]$, and $\int _{-\infty }^{\infty }\!\phi _{\lambda }(x)\,\mathrm {d}x=1$. Using substitution, we find $\forall \,\lambda >0$:

$$\begin{aligned} \int _{-\infty }^{\infty }g\left( \int _{-\infty }^{x}\phi _{\lambda }(s)\,\mathrm {d}s\right) \,\phi _{\lambda }(x)\,\mathrm {d}x=\int _{0}^{1}g(s)\mathrm {d}s. \end{aligned}$$

Let $\epsilon >0$, then by continuity of f(x) at $x=0$ there $\exists \;\bar{\lambda }$ such that if $x\in [-\bar{\lambda },\bar{\lambda }]$, then

$$\begin{aligned} |f(x)-f(0)|<\frac{\epsilon }{\max \left( 1,\int _{0}^{1}g(s)\mathrm {d}s\right) }. \end{aligned}$$

We note:

$$\begin{aligned} \begin{aligned}&\left| \,\int _{-\infty }^{\infty }g\left( \int _{-\infty }^{x}\delta (s)\mathrm {d}s\right) \,f(x)\,\delta (x)\,\mathrm {d}x-f(0)\,\int _{0}^{1}g(x)\mathrm {d}x\,\right| \nonumber \\&\quad =\lim _{\lambda {\rightarrow }0^{+}}\left| \,\int _{-\infty }^{\infty }g\left( \int _{-\infty }^{x}\phi _{\lambda }(s)\mathrm {d}s\right) \,f(x)\,\phi _{\lambda }(x)\,\mathrm {d}x\right. \\&\left. \qquad -f(0)\,\int _{-\infty }^{\infty }g\left( \int _{-\infty }^{x}\phi _{\lambda }(s)\mathrm {d}s\right) \,\phi _{\lambda }(x)\,\mathrm {d}x\,\right| \\&\quad \le \lim _{\lambda {\rightarrow }0^{+}}\,\int _{-\infty }^{\infty }\big |f(x)-f(0)\big |\,g\!\left( \int _{-\infty }^{x}\phi _{\lambda }(s)\mathrm {d}s\right) \,\phi _{\lambda }(x)\,\mathrm {d}x. \end{aligned} \end{aligned}$$

We note if $0<\lambda <\bar{\lambda }$, then:

$$\begin{aligned}&\int _{-\infty }^{\infty }\big |f(x)-f(0)\big |\,g\!\left( \int _{-\infty }^{x}\phi _{\lambda }(s)\mathrm {d}s\right) \,\phi _{\lambda }(x)\,\mathrm {d}x\nonumber \\&\quad \le \frac{\epsilon }{\max \left( 1,\int _{0}^{1}g(s)\mathrm {d}s\right) }\int _{-\infty }^{\infty }g\!\left( \int _{-\infty }^{x}\phi _{\lambda }(s)\mathrm {d}s\right) \,\phi _{\lambda }(x)\,\mathrm {d}x\nonumber \\&\quad =\,\epsilon . \end{aligned}$$

Since $\epsilon$ can be chosen to be arbitrarily small, we finally see:

$$\begin{aligned}&\lim _{\lambda {\rightarrow }0^{+}}\left| \,\int _{-\infty }^{\infty }g\left( \int _{-\infty }^{x}\phi _{\lambda }(s)\mathrm {d}s\right) \,f(x)\,\phi _{\lambda }(x)\,\mathrm {d}x-f(0)\right. \\&\left. \quad \int _{-\infty }^{\infty }g\left( \int _{-\infty }^{x}\phi _{\lambda }(s)\mathrm {d}s\right) \,\phi _{\lambda }(x)\,\mathrm {d}x\,\right| =0. \end{aligned}$$

thus concluding the proof of the Lemma.

We define $g(t)=\delta (t)$, $r(t)=\alpha _2e^{-\nu _e\tau }\,\delta (t-t_2)$, $m_1(t)=\gamma _1\,\delta (t-t_{l_1})$, $m_2(t)=\gamma _2\,\delta (t-t_{l_2})$. We further assume $t_o+\tau<t_{l_1}<t_{l_2}<T$, if this condition is not held it easy to see that the population will face extinction as the larvae will never have an opportunity to pupate. It follows from Eq. (4) that:

$$\begin{aligned}&\Lambda (\mu )=\alpha _1\alpha _2\,e^{-\nu _e\tau }\,\int ^T_\tau \mathrm {d}s_4\int ^{s_4}_\tau \mathrm {d}s_3\int ^{s_3}_\tau \mathrm {d}s_2\int ^{s_2-\tau }_0\mathrm {d}s_1\;\delta (s_2-t_o-\tau )\,\delta (s_1)\\&\quad \exp \!\bigg (\mu ^2\big (d_{l_2}(s_4-s_3)+d_{l_1}(s_3-s_2)+d_a(s_2-\tau -s_1)\big )\bigg )\\&\quad \times \exp \!\big (-\nu _p(T-s_4)-\nu _{l_2}(s_4-s_3)-\nu _{l_1}(s_3-s_2)-\nu _a(s_2-\tau -s_1)\big )\\&\quad \gamma _2\,\delta (s_4-t_{l_2})\,\exp \!\left( -\gamma _2\int ^{s_4}_{s_3}\mathrm {d}z\,\delta (z-t_{l_2})\right) \\&\quad \gamma _1\,\delta (s_3-t_{l_1})\,\exp \!\left( -\gamma _1\int ^{s_3}_{s_2}\mathrm {d}z\,\delta (z-t_{l_1})\right) . \end{aligned}$$

The $\mathrm {d}s_1,\,\mathrm {d}s_2$ integrals are straightforward, with $s_1,\,s_2$ being replaced by $0,\,t_o+\tau$, respectively, and replacing the lower bounds of the $\mathrm {d}s_4,\,\mathrm {d}s_3$ integrals with $t_o+\tau$. The $\mathrm {d}s_4$ and $\mathrm {d}s_3$ integral are more complicated and require subsequent applications of Lemma 4.1, with the $g(\cdot )$ of Lemma 4.1 being identified as $\exp (-\gamma _1\,\varvec{\cdot }),\;\exp (-\gamma _2\,\varvec{\cdot })$ respectively.

The moment generator thus becomes

$$\begin{aligned} \Lambda (\mu )&=\alpha _1\alpha _2\,e^{-\nu _e\tau }\;\big (1-e^{-\gamma _1}\big )\,\big (1-e^{-\gamma _2}\big )\\&\quad \times \exp \!\bigg (\mu ^2\big (d_{l_2}(t_{l_2}-t_{l_1})+d_{l_1}(t_{l_1}-t_o-\tau )+d_at_o\big )\bigg )\\&\quad \times \exp \!\big (-\nu _p(T-t_{l_2})-\nu _{l_2}(t_{l_2}-t_{l_1})-\nu _{l_1}(t_{l_1}-t_o-\tau )-\nu _at_o\big ). \end{aligned}$$

The condition of positive spread speed, $\Lambda (0)>1$, is equivalent to

$$\begin{aligned} \alpha _1\alpha _2>\frac{\,\exp \!\Big (\nu _p(T-t_{l_2})+\nu _{l_2}(t_{l_2}-t_{l_1})+\nu _{l_1}(t_{l_1}-t_o-\tau )+\nu _e\tau +\nu _at_o\Big )\,}{(1-e^{-\gamma _1})(1-e^{-\gamma _2})}. \end{aligned}$$

Taking the natural log of $\Lambda (\mu )$, we find

$$\begin{aligned} \ln \big (\Lambda (\mu )\big )&=\ln \!\bigg (\alpha _1\alpha _2\,e^{-\nu _e\tau }\;\big (1-e^{-\gamma _1}\big )\,\big (1-e^{-\gamma _2}\big )\bigg )\\&\quad -\nu _p(T-t_{l_2})-\nu _{l_2}(t_{l_2}-t_{l_1})-\nu _{l_1}(t_{l_1}-t_o-\tau )-\nu _at_o\\&\quad +\mu ^2\bigg (d_{l_2}(t_{l_2}-t_{l_1})+d_{l_1}(t_{l_1}-t_o-\tau )+d_at_o\bigg ). \end{aligned}$$

We can thus rewrite $\frac{\ln (\Lambda (\mu ))}{\mu }$ as

$$\begin{aligned} \frac{A+\mu ^2B}{\mu } \end{aligned}$$

where $A,\,B$ are defined by

$$\begin{aligned} A:&=\ln \!\bigg (\alpha _1\alpha _2\,e^{-\nu _e\tau }\;\big (1-e^{-\gamma _1}\big )\,\big (1-e^{-\gamma _2}\big )\bigg )-\nu _p(T-t_{l_2})\\&\quad -\nu _{l_2}(t_{l_2}-t_{l_1})-\nu _{l_1}(t_{l_1}-t_o-\tau )-\nu _at_o\\ B:&=d_{l_2}(t_{l_2}-t_{l_1})+d_{l_1}(t_{l_1}-t_o-\tau )+d_at_o. \end{aligned}$$

If $\Lambda (0)>1$, then $A>0$, while B is always positive. It then follows from elementary calculus that

$$\begin{aligned} \inf _{\mu >0}\,\frac{A+\mu ^2B}{\mu }=2\sqrt{AB\;}. \end{aligned}$$

Therefore,

$$\begin{aligned} c^*&=2\sqrt{AB\;}\\&=2\sqrt{d_{l_2}(t_{l_2}-t_{l_1})+d_{l_1}(t_{l_1}-t_o-\tau )+d_at_o\;}\;\\&\quad \times \sqrt{\ln \!\bigg (\alpha _1\alpha _2\,e^{-\nu _e\tau }\,(1-e^{-\gamma _1})(1-e^{-\gamma _2})\bigg )-\nu _p(T-t_{l_2})-\nu _{l_2}(t_{l_2}-t_{l_1})-\nu _{l_1}(t_{l_1}-t_o-\tau )-\nu _e\tau -\nu _at_o\ }\,. \end{aligned}$$

$\square$

Appendix D: Numerical Validation of Results in Sect. 3.2

As a further validation of the analytic result for $c^*$ in Eq. (5), we numerically integrate equations in model (1), starting with a smooth compact piecewise polynomial distribution. For subsequent years, we generate the end of the year pupa population densities $P_n(x)$ function. We then define $\epsilon$ to be some small fraction of the equilibrium population, and for year n we define $x_n>0$ such that $P_n(x_n)=\epsilon$. Taking a linear least square fit to the tuples $(n,x_n)$ and rejecting $(0,x_0)$ through $(5,x_5)$ so as to not include transients, we are able to numerically approximate the spreading speed as the slope of the fitted line. We summarize the results in Fig. 14. The value of the spreading speed obtained from the slope of fitted line is 1.06, a $6\%$ relative error of the analytically determined speed 1 given by Eq. (5).

Fig. 14

Wave propagation and spreading speed. Parameter values are chosen as $d_a = 1,\ T= 1.23,\ t_o = 0.23,\ m_1=m_2=3,\ \nu _a=\nu _e=\nu _{l_1} =\nu _{l_2}=1,\ \nu _p=0.5,\ \alpha _1 =0.8,\ \alpha _2 =12.5,\ \beta _a =0.4,\ \beta _{l_1}=\beta _{l_2}=0.06,\ \tau =.01$ and $\epsilon =5$. The slope of the fitted line is 1.06, the calculated speed is 1.00. We see on panel b that the position of the threshold population is nearly linear with time. a Wave fronts for even years, b $(n, x_n)$ and fitted line (Color figure online)