An Alternative Formulation for a Distributed Delayed Logistic Equation

Abstract

We study an alternative single species logistic distributed delay differential equation (DDE) with decay-consistent delay in growth. Population oscillation is rarely observed in nature, in contrast to the outcomes of the classical logistic DDE. In the alternative discrete delay model proposed by Arino et al. (J Theor Biol 241(1):109–119, 2006), oscillatory behavior is excluded. This study adapts their idea of the decay-consistent delay and generalizes their model. We establish a threshold for survival and extinction: In the former case, it is confirmed using Lyapunov functionals that the population approaches the delay modified carrying capacity; in the later case the extinction is proved by the fluctuation lemma. We further use adaptive dynamics to conclude that the evolutionary trend is to make the mean delay in growth as short as possible. This confirms Hutchinson’s conjecture (Hutchinson in Ann N Y Acad Sci 50(4):221–246, 1948) and fits biological evidence.

Appendices

Appendix

Dependence of ${\bar{N}}$ on Parameters $\mu$ and $\tau$

Lemma 3

${\bar{N}}$ is a decreasing function of $\mu$.

Proof

From Eq. (13), the positive equilibrium occurs at the intersection of $f_1(x)$ and $f_2(x)$, defined in (14). Considering ${\bar{N}}$ as a function of $\mu$, denote $f_1(x,\mu )$ and $f_2(x,\mu )$ as before. Note that, when $\mu _1<\mu _2$, $f_1(x,\mu _1)$ and $f_1(x,\mu _2)$ do not intersect, and $f_2(x,\mu _1)$ only intersects $f_2(x,\mu _2)$ at $x=0$. Because $f_1(0,\mu _1)>f_1(0,\mu _2)$, it follows that $f_1(x,\mu _1)>f_1(x,\mu _2)$ and so $f_1$ is decreasing in $\mu$. From the fact that $f_2$ is increasing in $\mu$, it follows that the intersection point ${\bar{N}}$ is a decreasing function of $\mu$. $\square$

Lemma 4

${\bar{N}}$ is a decreasing function of the mean delay $\tau$ for the kernel satisfying (10).

Proof

Consider ${\bar{N}}$ as a function of $\tau$ and denote the partial derivative with respective to $\tau$ by $\partial _{\tau }$. Then from (13)

$$\begin{aligned} \gamma \int _0^{\infty }\frac{\mu {\text{ e }}^{-\mu s} \partial _{\tau } k(s,\tau )}{\mu +\kappa (1-{\text{ e }}^{-\mu s}){\bar{N}}}\text {d}s-\gamma \int _0^{\infty }\frac{\mu {\text{ e }}^{-\mu s} k(s,\tau )\kappa (1-{\text{ e }}^{-\mu s}){\bar{N}}'(\tau )}{[\mu +\kappa (1-{\text{ e }}^{-\mu s}){\bar{N}}]^2}\text {d}s =\kappa {\bar{N}}'(\tau ) \end{aligned}$$

or

$$\begin{aligned} {\bar{N}}'(\tau )=\frac{\gamma \int _0^{\infty }\frac{\mu {\text{ e }}^{-\mu s} \partial _{\tau } k(s,\tau )}{\mu +\kappa (1-{\text{ e }}^{-\mu s}){\bar{N}}}\text {d}s}{\kappa \left( 1+\gamma \int _0^{\infty }\frac{\mu {\text{ e }}^{-\mu s} k(s,\tau )(1-{\text{ e }}^{-\mu s})}{[\mu +\kappa (1-{\text{ e }}^{-\mu s}){\bar{N}}]^2}\text {d}s \right) }. \end{aligned}$$

If k is the kernel for the model with the gamma distribution, then $\partial _{\tau }k(s,\tau )<0$ and so ${\bar{N}}'(\tau )<0$.

If k is the kernel for the model with the uniform distribution, then $k(s,\tau )=\frac{\rho }{\tau }\chi _{[\tau (1-\frac{1}{2\rho }),\tau (1+\frac{1}{2\rho })]}$, where $\chi _I(s)$ is 1 if $s\in I$ and is 0 otherwise. Using the change of variables $u=\frac{s}{\tau }$,

$$\begin{aligned}\begin{aligned}&\partial _{\tau }\displaystyle \int _0^{\infty }\frac{\mu {\text{ e }}^{-\mu s} k(s,\tau )}{\mu +\kappa (1- {\text{ e }}^{-\mu s}){\bar{N}}}\text {d}s\\&\quad = \, \partial _{\tau }\displaystyle \int _{1-\frac{1}{2\rho }}^{1+\frac{1}{2\rho }}\frac{\mu {\text{ e }}^{-\mu \tau u} \rho }{\mu +\kappa (1-{\text{ e }}^{-\mu \tau u}){\bar{N}}}\text {d}u\\&\quad =\displaystyle \int _{1-\frac{1}{2\rho }}^{1+\frac{1}{2\rho }}\frac{1}{(\mu +\kappa (1-{\text{ e }}^{-\mu \tau u}){\bar{N}})^2}[ -\mu ^3u\rho {\text{ e }}^{-\mu \tau u}\\&\qquad -\,\mu \rho {\text{ e }}^{-\mu \tau u}\kappa (1-{\text{ e }}^{-\mu \tau u}){\bar{N}}' ]\text {d}u=\kappa {\bar{N}}'(\tau ). \end{aligned} \end{aligned}$$

Thus we have ${\bar{N}}'(\tau )<0$.

If k is the kernel for the model with the tent distribution, then $k(s,\tau )=\frac{s}{\tau ^2}\chi _{[0,\tau ]}+\frac{2\tau -s}{\tau ^2}\chi _{[\tau ,2\tau ]}$. Let $u=\frac{s}{\tau }$,

$$\begin{aligned} \begin{aligned}&\partial _{\tau }\displaystyle \int _0^{\infty }\frac{\mu {\text{ e }}^{-\mu s} k(s,\tau )}{\mu +\kappa (1- {\text{ e }}^{-\mu s}){\bar{N}}}\text {d}s\\&\quad = \partial _{\tau }\displaystyle \int _{0}^{1}\frac{\mu {\text{ e }}^{-\mu \tau u} u}{\mu +\kappa (1-{\text{ e }}^{-\mu \tau u}){\bar{N}}}\text {d}u+\partial _{\tau }\displaystyle \int _{1}^{2}\frac{\mu {\text{ e }}^{-\mu \tau u} (1-u)}{\mu +\kappa (1-{\text{ e }}^{-\mu \tau u}){\bar{N}}}\text {d}u\\&\quad = \displaystyle \int _{0}^{1}\frac{1}{(\mu +\kappa (1-{\text{ e }}^{-\mu \tau u}){\bar{N}})^2}[ -\mu ^3u^2 {\text{ e }}^{-\mu \tau u}-\mu u {\text{ e }}^{-\mu \tau u}\kappa (1-{\text{ e }}^{-\mu \tau u}){\bar{N}}' ]\text {d}u\\&\qquad +\displaystyle \int _{1}^{2}\frac{1}{(\mu +\kappa (1-{\text{ e }}^{-\mu \tau u}){\bar{N}})^2}[ -\mu ^3u(1-u)\rho {\text{ e }}^{-\mu \tau u}\\&\qquad -\,\mu (1-u) {\text{ e }}^{-\mu \tau u}\kappa (1-{\text{ e }}^{-\mu \tau u}){\bar{N}}' ]\text {d}u\\&\quad =\kappa {\bar{N}}'(\tau ) . \end{aligned} \end{aligned}$$

Thus we have ${\bar{N}}'(\tau )<0$. $\square$

Appendix

Invasion Exponent

If k is the kernel for the model with the uniform distribution (12b), then the invasion exponent is

$$\begin{aligned} S_{\tau _1}(\tau _2)=\frac{\gamma }{\kappa {\bar{N}}(\tau _1)}\frac{\rho }{\tau _2}\ln \left( \frac{\mu +\kappa (1-{\text{ e }}^{-\mu \tau _2(1+\frac{1}{2\rho })}){\bar{N}}(\tau _1)}{\mu +\kappa (1-{\text{ e }}^{-\mu \tau _2(1-\frac{1}{2\rho })}){\bar{N}}(\tau _1)} \right) -\left( \mu +\kappa {\bar{N}}(\tau _1)\right) . \end{aligned}$$

If k is the kernel for the model with the tent distribution (12d), then using Maple (2009), the invasion exponent $S_{\tau _1}(\tau _2)$ is

$$\begin{aligned}&\frac{\gamma }{\tau _2^2\mu ^2\kappa ^2{\bar{N}}(\tau _1)}\left[ 2\tau _2\mu \ln \left( -\kappa {\bar{N}}(\tau _1){{\text{ e }}^{2\tau _2\mu }}-\mu {{\text{ e }}^{2 \tau _2\mu }}+\kappa {\bar{N}}(\tau _1) \right) \right. \\&\quad \left. -\,2\tau _2\mu \ln \left( {\frac{\kappa {\bar{N}}(\tau _1){{\text{ e }} ^{2\tau _2\mu }}+\mu {{\text{ e }}^{2\tau _2\mu }}-\kappa {\bar{N}}(\tau _1)}{\kappa {\bar{N}}(\tau _1)+\mu }} \right) \right. \\&\quad \left. -\,2 \tau _2\mu \ln \left( -\kappa {\bar{N}}(\tau _1){{\text{ e }}^{\tau _2\mu }}+\kappa {\bar{N}}(\tau _1)-\mu {{\text{ e }}^{ \tau _2\mu }} \right) \right. \\&\quad \left. +\,2\tau _2\mu \ln \left( {\frac{\kappa {\bar{N}}(\tau _1){{\text{ e }}^{ \tau _2\mu }}-\kappa {\bar{N}}(\tau _1)+\mu {{\text{ e }}^{\tau _2\mu }}}{\kappa {\bar{N}}(\tau _1)+\mu }} \right) \right. \\&\quad \left. +\,{ dilog} \left( -{\frac{\kappa {\bar{N}}(\tau _1){{\text{ e }}^{-2\tau _2\mu }}-\kappa {\bar{N}}(\tau _1)-\mu }{\kappa {\bar{N}}(\tau _1)+\mu }} \right) \right. \\&\quad \left. -\,2{ dilog} \left( -{\frac{\kappa {\bar{N}}(\tau _1){{\text{ e }}^{-\tau _2\mu }}-\kappa {\bar{N}}(\tau _1)- \mu }{\kappa {\bar{N}}(\tau _1)+\mu }} \right) \right. \\&\quad \left. +\,{ dilog} \left( {\frac{\mu }{\kappa {\bar{N}}(\tau _1)+\mu }} \right) \right] , \end{aligned}$$

where

$$\begin{aligned} { dilog} \left( x \right) =\int _1^x\frac{\ln t}{1-t}\text {d}t . \end{aligned}$$