On first-order hyperbolic partial differential equations with two internal variables modeling population dynamics of two physiological structures

Abstract

In this paper we develop fundamental theories for a scalar first-order hyperbolic partial differential equation with two internal variables which models single-species population dynamics with two physiological structures such as age–age, age–maturation, age–size, and age–stage. Classical techniques of treating structured models with a single internal variable are generalized to study the double physiologically structured model. First, the semigroup is defined based on the solutions and its infinitesimal generator is determined. Then, the compactness of solution trajectories is established. Finally, spectrum theory is employed to investigate stability of the zero steady state and asynchronous exponential growth of solutions is studied when the zero steady state is unstable.

Appendix

In this Appendix, we prove some statements that were used in Sect. 4.

Proposition A.1

If $a^+=\infty$, ${\underline{\mu }}>0$, and Assumption 2.1holds, then $S_1(t)$ satisfies the hypothesis (ii) and $S_2(t), S_3(t)$ satisfy the hypothesis (iii) of Proposition 4.2.

Proof

We only need to show that $S_2(t)$ is compact for $t>0$, which is equivalent to show that for a bounded set K of E,

$$\begin{aligned}&\lim_{h\rightarrow 0, \, k\rightarrow 0} \int _{0}^{\infty } \int _{0}^{\infty }\big |S_2(t)\phi (a+h, a'+k)-S_2(t)\phi (a,a')\big |{\mathrm{d}}a{\mathrm{d}}a'=0 \end{aligned}$$

(A.1)

$$\begin{aligned} &\lim _{ h\rightarrow \infty, k\rightarrow \infty}\int _{h}^{\infty } \int _{k}^{\infty }\big |S_2(t)\phi (a,a')\big |{\mathrm{d}}a{\mathrm{d}}a'=0 \end{aligned}$$

(A.2)

uniformly for $\phi \in K$ (which can be found in [14, Theorem 21, p. 301]). Without loss of generality, assume $k>h$ and $h,k\rightarrow 0^+$, we have

$$\begin{aligned}&\int _{0}^{\infty } \int _{0}^{\infty }\big |S_2(t)\phi (a+h, a'+k)-S_2(t)\phi (a,a')\big |{\mathrm{d}}a'{\mathrm{d}}a\nonumber \\&\quad \le \underbrace{\int _{0}^{t-h} \int _{a}^{\infty }\big |S_2(t)\phi (a+h, a'+k)-S_2(t)\phi (a,a')\big |{\mathrm{d}}a'{\mathrm{d}}a}_{{{\text {region I}}}}\nonumber \\&\qquad +\underbrace{\int _{k-h}^{t-h} \int _{a+h-k}^{a}\big |S_2(t)\phi (a+h,a'+k)\big |{\mathrm{d}}a'{\mathrm{d}}a}_{{{\text {region II}}}}\nonumber \\&\qquad + \underbrace{\int _0^{k-h} \int _0^{a}\big |S_2(t)\phi (a+h,a'+k)\big | {\mathrm{d}}a'{\mathrm{d}}a}_{{{\text {region III}}}}+\underbrace{\int _{t-h}^{t} \int _a^{\infty }\big |S_2(t)\phi (a,a')\big | {\mathrm{d}}a'{\mathrm{d}}a}_{{{\text {region IV}}}}, \end{aligned}$$

(A.3)

as illustrated in Fig. 3b: $S_2(t)\phi (a, a')$ is non-trivial for points $(a,a')$ in regions I and IV, and $S_2(t)\phi (a+h, a'+k)$ is non-trivial for points $(a,a')$ in regions I, II, and III.

We first show

$$\begin{aligned}&\int _{0}^{t-h} \int _{a}^{\infty }\big |S_2(t)\phi (a+h, a'+k)-S_2(t)\phi (a,a')\big |{\mathrm{d}}a'{\mathrm{d}}a\\&\quad \le \int _{0}^{t} \int _{a}^{\infty }\big |b_\phi (t-a-h, a'+k-a-h)-b_\phi (t-a, a'-a)\big |e^{-\int _{0}^{a+h}\mu (s, s+a'+k-a-h){\mathrm{d}}s}{\mathrm{d}}a'{\mathrm{d}}a\\&\qquad +\int _{0}^{t-h} \int _{a}^{\infty }\big |b_\phi (t-a, a'-a)[e^{-\int _{0}^{a+h}\mu (s, s+a'+k-a-h){\mathrm{d}}s}-e^{-\int _{0}^{a}\mu (s, s+a'-a){\mathrm{d}}s}]\big |{\mathrm{d}}a'{\mathrm{d}}a \\&\quad :={\mathrm{I}}+{\mathrm{II}}, \end{aligned}$$

where

$$\begin{aligned} {\mathrm{II}}&\le \int _{0}^{t-h}\int _{a}^{\infty }b_\phi (t-a, a'-a)\big |e^{-\int _0^{a+h}\mu (s, s+a'-a){\mathrm{d}}s}[1-e^{\int _0^{a+h}\mu (s, s+a'-a)-\mu (s, s+a'+k-a-h){\mathrm{d}}s}]\\&\quad +e^{-\int _0^{a}\mu (s, s+a'-a){\mathrm{d}}s}[1-e^{-\int _{a}^{a+h}\mu (s, s+a'-a){\mathrm{d}}s}]\big |{\mathrm{d}}a'{\mathrm{d}}a\\&\le \int _{0}^{t-h}\int _{a}^{\infty }b_\phi (t-a, a'-a)\big (\max \{1-e^{-K_{\mu }(k-h)t},e^{K_{\mu }(k-h)t}-1\}+(1-e^{-{\bar{\mu }}h})\big ){\mathrm{d}}a'{\mathrm{d}}a\\&\le \big (\max \{1-e^{-K_{\mu }(k-h)t},e^{K_{\mu }(k-h)t}-1\}+(1-e^{-{\bar{\mu }}h})\big )\int _{0}^{t}\int _{0}^{\infty }b_\phi (t-a, s){\mathrm{d}}s{\mathrm{d}}a\\&\le 2\beta _{\max }\Vert \phi \Vert _E \big (\max \{1-e^{-K_{\mu }(k-h)t},e^{K_{\mu }(k-h)t}-1\}+(1-e^{-{\bar{\mu }}h})\big ) \int _{0}^{t}e^{4\beta _{\max }(t-a)}{\mathrm{d}}a \end{aligned}$$

based on our prior estimate in Sect. 2.1 and with $K_{\mu }$ being the Lipschitz constant for $\mu$, thus ${\mathrm{II}}\rightarrow 0$ uniformly for $\phi \in K$ as $h,k\rightarrow 0^+$. Next, we need to show that

$$\begin{aligned} \lim_{h\rightarrow 0, \, k\rightarrow 0} \int _{0}^{t}\int _{a}^{\infty }|b_\phi (t-a-h, a'+k-a-h)-b_\phi (t-a, a'-a)|{\mathrm{d}}a'{\mathrm{d}}a=0. \end{aligned}$$

(A.4)

Now by an alternative version of (2.7) and (2.8), we have

$$\begin{aligned}&\int _{0}^{t}\int _{a}^{\infty }|b_\phi (t-a-h, a'+k-a-h)-b_\phi (t-a, a'-a)|{\mathrm{d}}a'{\mathrm{d}}a\\&\quad \le \int _{0}^{t}\int _{a}^{\infty }\big |\int _{0}^{t-a-h}\int _{0}^{\infty }f_1(a'+k-a-h, t-a-h-p, s+t-a-h-p)b_\phi (p, s){\mathrm{d}}s{\mathrm{d}}p\\&\qquad -\int _{0}^{t-a}\int _{0}^{\infty }f_1(a'-a, t-a-p, s+t-a-p)b_\phi (p, s){\mathrm{d}}s{\mathrm{d}}p\big |{\mathrm{d}}a'{\mathrm{d}}a\\&\qquad +\int _{0}^{t}\int _{a}^{\infty }\big |\int _{0}^{t-a-h}\int _{0}^{\infty }g_1(a'+k-a-h, p+t-a-h-s, t-a-h-s)b'_\phi (s, p){\mathrm{d}}p{\mathrm{d}}s\\&\qquad -\int _{0}^{t-a}\int _{0}^{\infty }g_1(a'-a, p+t-a-s, t-a-s)b'_\phi (s, p){\mathrm{d}}p{\mathrm{d}}s\big |{\mathrm{d}}a'{\mathrm{d}}a\\&\qquad +\int _{0}^{t}\int _{a}^{\infty }\big |\int _{0}^{\infty }\int _{0}^{\infty }h_1(a'+k-a-h, p, s, t-a-h)\phi (p, s){\mathrm{d}}p{\mathrm{d}}s\\&\qquad -\int _{0}^{\infty }\int _{0}^{\infty }h_1(a'-a, p, s, t-a)\phi (p, s){\mathrm{d}}p{\mathrm{d}}s\big |{\mathrm{d}}a'{\mathrm{d}}a\\&\quad := {\mathrm{J}}_1+{\mathrm{J}}_2+{\mathrm{J}}_3, \end{aligned}$$

where

$$\begin{aligned} {\mathrm{J}_1}&\le \int _{0}^{t} \int _{a}^{\infty } \big (\int _{0}^{t-a-h}\int _{0}^{\infty }|f_1(a'+k-a-h, t-a-h-p, s+t-a-h-p)\\&\quad -f_1(a'-a, t-a-p, s+t-a-p)|b_\phi (p, s){\mathrm{d}}s{\mathrm{d}}p\\&\quad +\int _{t-a}^{t-a-h}\int _{0}^{\infty }f_1(a'-a, t-a-p, s+t-a-p)b_\phi (p, s){\mathrm{d}}s{\mathrm{d}}p\big ){\mathrm{d}}a'{\mathrm{d}}a\\&:={\mathrm{J}}_1^1+{\mathrm{J}}_1^2, \end{aligned}$$

in which

$$\begin{aligned}&|f_1(a'+k-a-h, t-a-h-p, s+t-a-h-p)-f_1(a'-a, t-a-p, s+t-a-p)|\\&\quad \le \beta _1(a'+k-a-h)\big |\beta _2(t-a-h-p, s+t-a-h-p)e^{-\int _{0}^{t-a-p-h}\mu (\sigma ,\sigma +s){\mathrm{d}}\sigma }\\&\qquad -\beta _2(t-a-p, s+t-a-p)e^{-\int _{0}^{t-a-p}\mu (\sigma ,\sigma +s){\mathrm{d}}\sigma }\big |\\&\qquad +|\beta _1(a'+k-a-h)-\beta _1(a'-a)|\beta _2(t-a-p, s+t-a-p)\\&\quad \le \beta _1(a'+k-a-h)\big |\beta _2(t-a-h-p, s+t-a-h-p)-\beta _2(t-a-p, s+t-a-p)\big |\\&\qquad +\beta _1(a'+k-a-h)\beta _2(t-a-p, s+t-a-p)\big |1-e^{-\int _{t-a-h-p}^{t-a-p}\mu (\sigma ,\sigma +s){\mathrm{d}}\sigma }\big |\\&\qquad +\big |\beta _1(a'+k-a-h)-\beta _1(a'-a)\big |\beta _2(t-a-p, s+t-a-p)\\&\quad \le \beta _1(a'+k-a-h)K_{\beta }2h+{\bar{\beta }}(a'+k-a-h)(1-e^{{\bar{\mu }}h})\\&\qquad +\big |\beta _1(a'+k-a-h)-\beta _1(a'-a)\big |\beta _2(t-a-p, s+t-a-p)\\&\quad \le \beta _1(a'+k-a-h)K_{\beta }2h+{\bar{\beta }}(a'+k-a-h)(1-e^{{\bar{\mu }}h})\\&\qquad +\big |\beta _1(a'+k-a-h)-\beta _1(a'-a)\big |\beta _2^{\sup } \end{aligned}$$

by Assumption 2.1(i) on $\beta$ being Lipschitz continuous and $K_{\beta }$ as the Lipschitz constant, where

$$\begin{aligned} \beta _2^{\sup }:=\sup _{(a, s)\in (0, \infty )\times (0, \infty )}\beta _2(a, s)<\infty \end{aligned}$$

because Assumption 2.1(iv) holds. Thus,

$$\begin{aligned} J_1^1&\le \int _{0}^{t}\int _{a}^{\infty }\big (\beta _1(a'+k-a-h)K_{\beta }2h+{\bar{\beta }}(a'+k-a-h)(1-e^{{\bar{\mu }}h})\big )\big (\int _{0}^{t-a-h}\int _{0}^{\infty }b_\phi (p, s){\mathrm{d}}s{\mathrm{d}}p\big ){\mathrm{d}}a'{\mathrm{d}}a\\&\quad +\int _{0}^{t}\int _{a}^{\infty }\big |\beta _1(a'+k-a-h)-\beta _1(a'-a)\big |\beta _2^{\sup }\big (\int _{0}^{t-a-h}\int _{0}^{\infty }b_\phi (p, s){\mathrm{d}}s{\mathrm{d}}p\big ){\mathrm{d}}a'{\mathrm{d}}a\\&\le \int _{0}^{t}\int _{a}^{\infty }\big (\beta _1(a'+k-a-h)K_{\beta }2h+{\bar{\beta }}(a'+k-a-h)(1-e^{{\bar{\mu }}h})\big )\big (\int _{0}^{t-a-h}2\beta _{\max }\Vert \phi \Vert _E e^{4\beta _{\max }p}{\mathrm{d}}p\big ){\mathrm{d}}a'{\mathrm{d}}a\\&\quad +\int _{0}^{t}\int _{a}^{\infty }\big |\beta _1(a'+k-a-h)-\beta _1(a'-a)\big |\beta _2^{\sup }\big (\int _{0}^{t-a-h}2\beta _{\max }\Vert \phi \Vert _E e^{4\beta _{\max }p}{\mathrm{d}}p\big ){\mathrm{d}}a'{\mathrm{d}}a\\&\rightarrow 0 \; {{\text { uniformly for }}} \phi \in K {{\text { as }}} h,k\rightarrow 0^+ \end{aligned}$$

and

$$\begin{aligned} J_1^2&\le \int _{0}^{t}\int _{a}^{\infty }{\bar{\beta }}(a'-a)\big (\int _{t-a}^{t-a-h}\int _{0}^{\infty }b_\phi (p, s){\mathrm{d}}s{\mathrm{d}}p\big ){\mathrm{d}}a'{\mathrm{d}}a\\&\le \big (\int _{0}^{\infty }{\bar{\beta }}(a'){\mathrm{d}}a'\big )\big (\int _{0}^{t}\int _{t-a-h}^{t-a}2\beta _{\max }\Vert \phi \Vert _E e^{4\beta _{\max }p}{\mathrm{d}}p{\mathrm{d}}a\big )\\&\rightarrow 0 \; {{\text { as }}} h,k\rightarrow 0^+ {{\text { uniformly for }}}\phi \in K, \end{aligned}$$

Therefore, we have $J_1\rightarrow 0$ as $h,k\rightarrow 0^+$ uniformly for $\phi \in K$. And the fact that $J_2\rightarrow 0$ uniformly for $\phi \in K$ as $h,k\rightarrow 0^+$ can be proved by using a similar argument. To show $J_3\rightarrow 0$, we first

$$\begin{aligned}&\big |h_1(a'+k-a-h,p,s,t-a-h)-h_1(a'-a,p,s,t-a)\big |\\&\quad \le \big |\beta (a'+k-a-h,p+t-a-h,s+t-a-h)-\beta (a'-a,p+t-a,s+t-a)\big |e^{-\int _0^{t-a-h}\mu (\sigma +p,\sigma +s){\mathrm{d}}\sigma }\\&\qquad +\beta (a'-a,p+t-a,s+t-a)e^{-\int _0^{t-a-h}\mu (\sigma +p,\sigma +s){\mathrm{d}}\sigma }\big |1-e^{-\int _{t-a-h}^{t-a}\mu (\sigma +p,\sigma +s){\mathrm{d}}\sigma }\big |\\&\quad \le \big |\beta (a'+k-a-h,p+t-a-h,s+t-a-h)-\beta (a'-a,p+t-a-h,s+t-a-h)\big |\\&\qquad +\beta _1(a'-a)\big |\beta _2(p+t-a-h,s+t-a-h)-\beta _2(p+t-a,s+t-a)\big |\\&\qquad +\beta (a'-a,p+t-a,s+t-a)(1-e^{-{\underline{\mu }}h})\\&\quad \le \big |\beta (a'+k-a-h,p+t-a-h,s+t-a-h)-\beta (a'-a,p+t-a-h,s+t-a-h)\big |\\&\qquad +\beta _1(a'-a)K_{\beta }2h+{\bar{\beta }}(a'-a)(1-e^{-{\underline{\mu }}h}). \end{aligned}$$

Then by applying Assumption  2.1(ii), we have $J_3\rightarrow 0$ uniformly for $\phi \in K$.

Therefore, we have the first term in (A.3) goes to 0 uniformly for $\phi \in K$. Secondly,

based on estimate (4.8) in the main text we have

$$\begin{aligned}&\int _{k-h}^{t-h} \int _{a+h-k}^{a}\big |S_2(t)\phi (a+h,a'+k)\big |{\mathrm{d}}a'{\mathrm{d}}a\\&\quad \le \int _{k-h}^{t-h} \int _{a+h-k}^{a}b_{\phi }(t-a-h,a'+k-a-h){\mathrm{d}}a'{\mathrm{d}}a\\&\quad \le \int _{k-h}^{t-h}\big |\int _{a+h-k}^{a}{\bar{\beta }}(a'){\mathrm{d}}a'\big |\big (4\Vert \phi \Vert _E\beta _{\max }\int _0^{t}e^{4\beta _{\max }p}{\mathrm{d}}pp+\Vert \phi \Vert _E\big ){\mathrm{d}}a\\&\quad \le \sup _{0<a<t}\big |\int _{a+h-k}^{a}{\bar{\beta }}(a'){\mathrm{d}}a'\big |\cdot t \cdot \big (4\Vert \phi \Vert _E\beta _{\max }\int _0^{t}e^{4\beta _{\max }p}{\mathrm{d}}p+\Vert \phi \Vert _E\big )\\&\quad \rightarrow 0 \; {{\text { as }}} h,k\rightarrow 0^+ {{\text { uniformly for }}}\phi \in K. \end{aligned}$$

Further,

$$\begin{aligned}&\int _0^{k-h} \int _0^{a}\big |S_2(t)\phi (a+h,a'+k)\big | {\mathrm{d}}a'{\mathrm{d}}a\\&\quad \le \int _0^{k-h} \int _0^{a} b_{\phi }(t-a-h,a'+k-a-h){\mathrm{d}}a'{\mathrm{d}}a\\&\quad \le \int _0^{k-h} \int _0^{\infty } b_{\phi }(t-a-h,s){\mathrm{d}}s{\mathrm{d}}a \le 2\beta _{\max }\Vert \phi \Vert _E\int _0^{k-h}e^{4\beta _{\max }(t-a-h)}{\mathrm{d}}a \\&\quad \rightarrow 0 \; {{\text { as }}} h,k\rightarrow 0^+ {{\text { uniformly for }}}\phi \in K. \end{aligned}$$

Lastly,

$$\begin{aligned}&\int _{t-h}^{t} \int _a^{\infty }\big |S_2(t)\phi (a,a')\big | {\mathrm{d}}a'{\mathrm{d}}a\\&\quad \le \int _{t-h}^{t} \int _a^{\infty } b_{\phi }(t-a,a'-a){\mathrm{d}}a'{\mathrm{d}}a\\&\quad \le \int _{t-h}^{t} \int _0^{\infty } b_{\phi }(t-a,s){\mathrm{d}}s{\mathrm{d}}a \le 2\beta _{\max }\Vert \phi \Vert _E\int _{t-h}^{t}e^{4\beta _{\max }(t-a)}{\mathrm{d}}a \\&\quad \rightarrow 0{{\text { as }}} h\rightarrow 0^+ {{\text { uniformly for }}}\phi \in K. \end{aligned}$$

We thus proved (A.1). For (A.2), we have

$$\begin{aligned}&\int _{h}^{\infty } \int _{k}^{\infty }\big |S_2(t)\phi (a,a')\big |{\mathrm{d}}a{\mathrm{d}}a'\\&\quad \le \int _{h}^{\infty } \int _0^{\infty }b_{\phi }(t-a,s){\mathrm{d}}s{\mathrm{d}}a \\&\quad \le 2\Vert \phi \Vert _E\beta _{\max }\int _{h}^{\infty }e^{4\beta _{\max }(t-a)}{\mathrm{d}}a\rightarrow 0 \end{aligned}$$

as $h,k\rightarrow 0$ uniformly for $\phi \in K$. We can show that $S_3(t)$ is compact for sufficiently large t in the same way. $\square$