# 具有阶段结构的非自治捕食–食饵模型Non-Autonomous Predator-Prey Model with Stage Structure

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This paper studies a kind of non-autonomous predator-prey model with stage structure and mod el parameters depended on time-varying coefficients. In the model it is assumed that the predator has stage structure, prey has exponential growth rate, and dynamics is analyzed qualitatively when all the parameters on the model parameters depend on time-varying coefficients. By defining the net reproductive number of the predator, a uniform persistence and extinction of the predators are obtained by using the comparison principle, respectively. Finally the correctness of the theory is verified by numerical simulation.

1. 引言

$\begin{array}{l}\frac{dx}{dy}=x\left(r-\frac{x}{k}\right)-\frac{\alpha xy}{a+bx+cy}\\ \frac{dy}{dy}=-dy+\frac{\beta xy}{a+bx+cy}\end{array}$ (1)

$\left\{\begin{array}{l}{x}^{\prime }\left(t\right)=rx\left(t\right)\left[1-\frac{x\left(t\right)}{K}\right]-\frac{bx\left(t\right)y\left(t\right)}{1+{k}_{1}x\left(t\right)+{k}_{\text{2}}y\left(t\right)}\\ {y}^{\prime }\left(t\right)=\frac{nbx\left(t-\tau \right)y\left(t-\tau \right){\text{e}}^{-{d}_{i}\tau }}{1+{k}_{1}x\left(t-\tau \right)+{k}_{2}y\left(t-\tau \right)}-dy\left(t\right)\\ {{y}^{\prime }}_{j}\left(t\right)=\frac{nbx\left(t\right)y\left(t\right)}{1+{k}_{1}x\left(t\right)+{k}_{\text{2}}y\left(t\right)}-\frac{nbx\left(t-\tau \right)y\left(t-\tau \right){\text{e}}^{-{d}_{i}\tau }}{1+{k}_{1}x\left(t-\tau \right)+{k}_{2}y\left(t-\tau \right)}-{d}_{j}{y}_{j}\left(t\right)\end{array}$ (2)

2. 模型的建立

$\left\{\begin{array}{l}{x}^{\prime }\left(t\right)=\lambda \left(t\right)-\mu \left(t\right)x\left(t\right)-\frac{b\left(t\right)x\left(t\right)y\left(t\right)}{1+{k}_{1}\left(t\right)x\left(t\right)+{k}_{\text{2}}\left(t\right)y\left(t\right)}\\ {y}^{\prime }\left(t\right)=\frac{n\left(t-\tau \right)b\left(t-\tau \right)x\left(t-\tau \right)y\left(t-\tau \right){\text{e}}^{-{\int }_{t-\tau }^{t}{d}_{i}\left(s\right)\text{d}s}}{1+{k}_{1}\left(t-\tau \right)x\left(t-\tau \right)+{k}_{2}\left(t-\tau \right)y\left(t-\tau \right)}-d\left(t\right)y\left(t\right)\\ {{y}^{\prime }}_{j}\left(t\right)=\frac{n\left(t\right)b\left(t\right)x\left(t\right)y\left(t\right)}{1+{k}_{1}\left(t\right)x\left(t\right)+{k}_{\text{2}}\left(t\right)y\left(t\right)}-\frac{n\left(t-\tau \right)b\left(t-\tau \right)x\left(t-\tau \right)y\left(t-\tau \right){\text{e}}^{-{\int }_{t-\tau }^{t}{d}_{i}\left(s\right)\text{d}s}}{1+{k}_{1}\left(t-\tau \right)x\left(t-\tau \right)+{k}_{2}\left(t-\tau \right)y\left(t-\tau \right)}-{d}_{j}{y}_{j}\left(t\right)\end{array}$ (3)

3. 准备工作

(A1) 函数 $\lambda \left(\cdot \right),\mu \left(\cdot \right),b\left(\cdot \right),{k}_{1}\left(\cdot \right),{k}_{2}\left(\cdot \right),n\left(\cdot \right),{d}_{j}\left(\cdot \right),d\left(\cdot \right)$$\left(0,+\infty \right)$ 上的正的有界函数且连续；

(A2) 若函数 $g\left(t\right)$$\left[0,+\infty \right)$ 上的连续有界函数，则

${g}_{\infty }=\underset{t\to \infty }{\mathrm{lim}}\mathrm{inf}g\left(t\right)$ , ${g}^{\infty }=\underset{t\to \infty }{\mathrm{lim}}\mathrm{sup}g\left(t\right)$ .

(A3) 存在正的常数 ${\omega }_{1},{\omega }_{2}>0$ ，使得

$\underset{t\to \infty }{\mathrm{lim}}\mathrm{inf}{\int }_{t}^{t+{\omega }_{1}}\lambda \left(s\right)\text{d}s>0$ , $\underset{t\to \infty }{\mathrm{lim}}\mathrm{inf}{\int }_{t}^{t+{\omega }_{2}}\mu \left(s\right)\text{d}s>0$

(A4) 对于所有 $t\ge 0$ 来说，都有 ${d}_{j}\left(t\right)>d\left(t\right)$

$\left\{\begin{array}{l}x\left(\theta \right)={\varphi }_{1}\left(\theta \right),\\ y\left(\theta \right)={\varphi }_{2}\left(\theta \right),\\ {y}_{j}\left(\theta \right)={\varphi }_{3}\left(\theta \right),\\ -\tau \le \theta \le 0,\\ {\varphi }_{i}\left(0\right)>0,i=1,2,3.\end{array}$ (4)

$‖\varphi ‖=\underset{-\tau \le \theta \le 0}{\mathrm{max}}\left\{|{\varphi }_{1}\left(\theta \right)|,|{\varphi }_{2}\left(\theta \right)|,|{\varphi }_{3}\left(\theta \right)|\right\}.$

$\left\{\begin{array}{l}q\le \underset{t\to +\infty }{\mathrm{lim}}\mathrm{inf}x\left(t\right)\le \underset{t\to +\infty }{\mathrm{lim}}\mathrm{sup}x\left(t\right)\le L,\\ {\stackrel{˜}{q}}_{1}\le \underset{t\to +\infty }{\mathrm{lim}}\mathrm{inf}y\left(t\right)\le \underset{t\to +\infty }{\mathrm{lim}}\mathrm{sup}y\left(t\right)\le {\stackrel{˜}{L}}_{1},\\ {\stackrel{˜}{q}}_{2}\le \underset{t\to +\infty }{\mathrm{lim}}\mathrm{inf}{y}_{j}\left(t\right)\le \underset{t\to +\infty }{\mathrm{lim}}\mathrm{sup}{y}_{j}\left(t\right)\le {\stackrel{˜}{L}}_{2}.\end{array}$

${z}^{\prime }\left(t\right)=\lambda \left(t\right)-\mu \left(t\right)z\left(t\right)$ (5)

1) 方程(5)具有初值 $z\left(0\right)>0$ 任意解的最终极限 ${z}^{*}\left(t\right)$${R}_{+}$ 内有界且全局一致吸引；

2) 存在常数m使得 $m\le \underset{t\to +\infty }{\mathrm{lim}}\mathrm{inf}z\left(t\right)\le \underset{t\to +\infty }{\mathrm{lim}}\mathrm{sup}z\left(t\right)\le M$

3) 若方程(5)是ω-周期的，则方程(5)存在唯一非负ω的周期解且全局一致吸引；

4) 对任意的 $t\ge 0$$\mu \left(t\right)>0$ ，若

$0<\underset{t\to \infty }{\mathrm{lim}}\mathrm{inf}\frac{\lambda \left(t\right)}{\mu \left(t\right)}\le \underset{t\to \infty }{\mathrm{lim}}\mathrm{sup}\frac{\lambda \left(t\right)}{\mu \left(t\right)}<\infty$

${\left(\frac{\lambda }{\mu }\right)}_{\infty }<\underset{t\to +\infty }{\mathrm{lim}}\mathrm{inf}z\left(t\right)\le \underset{t\to +\infty }{\mathrm{lim}}\mathrm{sup}z\left(t\right)<{\left(\frac{\lambda }{\mu }\right)}^{\infty }$

${\left(\frac{\lambda }{\mu }\right)}_{\infty }=\underset{t\to \infty }{\mathrm{lim}}\mathrm{inf}\frac{\lambda \left(t\right)}{\mu \left(t\right)},\text{\hspace{0.17em}}{\left(\frac{\lambda }{\mu }\right)}^{\infty }=\underset{t\to \infty }{\mathrm{lim}}\mathrm{sup}\frac{\lambda \left(t\right)}{\mu \left(t\right)}$

$x\left(t\right)=x\left(0\right){\text{e}}^{-{\int }_{0}^{t}\left[\mu \left(s\right)+\frac{n\left(s\right)y\left(s\right)}{1+{k}_{1}\left(s\right)x\left(s\right)+{k}_{2}\left(s\right)y\left(s\right)}\right]\text{d}s}+{\int }_{0}^{t}\lambda \left(s\right){\text{e}}^{{\int }_{t}^{s}\left(\mu \left(\theta \right)+\frac{n\left(\theta \right)y\left(\theta \right)}{1+{k}_{1}\left(\theta \right)x\left(\theta \right)+{k}_{2}\left(\theta \right)y\left(\theta \right)}\right)\text{d}s}$

$y\left(t\right)\ge y\left(0\right){\text{e}}^{-{\int }_{0}^{t}d\left(s\right)\text{d}s}$

$\begin{array}{c}{y}_{j}\left(t\right)={\int }_{t-\tau }^{t}\frac{n\left(s\right)b\left(s\right)x\left(s\right)y\left(s\right){\text{e}}^{-{\int }_{s}^{t}{d}_{j}\left(u\right)\text{d}u}}{1+{k}_{1}\left(s\right)x\left(s\right)+{k}_{2}\left(s\right)y\left(s\right)}\text{d}s\\ ={\int }_{t-\tau }^{0}\frac{n\left(s\right)b\left(s\right)x\left(s\right)y\left(s\right){\text{e}}^{-{\int }_{s}^{t}{d}_{j}\left(u\right)\text{d}u}}{1+{k}_{1}\left(s\right)x\left(s\right)+{k}_{2}\left(s\right)y\left(s\right)}\text{d}s+{\int }_{0}^{t}\frac{n\left(s\right)b\left(s\right)x\left(s\right)y\left(s\right){\text{e}}^{-{\int }_{s}^{t}{d}_{j}\left(u\right)\text{d}u}}{1+{k}_{1}\left(s\right)x\left(s\right)+{k}_{2}\left(s\right)y\left(s\right)}\text{d}s\end{array}$ (6)

$x\left(0\right)>0$ 则对任意的 $t\ge 0$ 总有 $x\left(t\right)>0$ ，显然对任意的 $t\in \left[0,\tau \right]$$y\left(t\right)>0$${y}_{j}\left(t\right)$ 对任意的 $t\in \left[\theta ,\tau \right]$${y}_{j}\left(t\right)>0$ 。从而得到了 $x\left(t\right),y\left(t\right),{y}_{j}\left(t\right)$$\left[-\tau ,\tau \right]$ 上的正性。因此当 $x\left(0\right),y\left(0\right)>0,{y}_{j}\left(0\right)>0$ 时,对于任意的 $t\ge 0$ ，总有 $x\left(t\right)>0,y\left(t\right)>0,{y}_{j}\left(t\right)>0$

$\eta \left(t\right)=x\left(t\right)+y\left(t\right)+{y}_{j}\left(t\right)$

$\begin{array}{c}{\eta }^{\prime }\left(t\right)=\lambda \left(t\right)-\frac{b\left(t\right)x\left(t\right)y\left(t\right)}{1+{k}_{1}\left(t\right)x\left(t\right)+{k}_{2}\left(t\right)y\left(t\right)}+\frac{n\left(t\right)b\left(t\right)x\left(t\right)y\left(t\right)}{1+{k}_{1}\left(t\right)x\left(t\right)+{k}_{2}\left(t\right)y\left(t\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-d\left(t\right)y\left(t\right)-{d}_{j}\left(t\right){y}_{j}\left(t\right)-\mu \left(t\right)x\left(t\right)\\ \le {\lambda }^{\infty }\left(\frac{{n}^{\infty }{b}^{\infty }}{{K}_{2}{}_{{}_{\infty }}}-{\mu }_{\infty }\right)x\left(t\right)-{d}_{j}{}_{{}_{\infty }}{y}_{j}\left(t\right)-{d}_{\infty }y\left(t\right)\end{array}$ (7)

$\beta =\mathrm{min}\left\{{d}_{j}{}_{{}_{\infty }},{d}_{\infty }\right\}$ ，由(7)式可得：

${\eta }^{\prime }\left(t\right)+\eta \left(t\right)\le {\lambda }^{\infty }+\beta x\left(t\right)+\left(\frac{{n}^{\infty }{b}^{\infty }}{{K}_{2}{}_{{}_{\infty }}}-{\mu }_{\infty }\right)x\left(t\right)$

$\underset{t\to +\infty }{\mathrm{lim}}\mathrm{sup}x\left(t\right)\le \frac{{\lambda }^{\infty }}{{\mu }_{\infty }}.$ (8)

${\eta }^{\prime }\left(t\right)+\beta \eta \left(t\right)\le {\lambda }^{\infty }+\left(\beta +\frac{{n}^{\infty }{b}^{\infty }}{{K}_{2}{}_{{}_{\infty }}}-{\mu }_{\infty }\right)\cdot \frac{{\lambda }^{\infty }}{{\mu }_{\infty }}:=M$ (9)

$\underset{t\to +\infty }{\mathrm{lim}}\mathrm{sup}\eta \left(t\right)<\frac{M}{\beta }:=\stackrel{¯}{L}$ (10)

$\Omega :=\left\{\left(x\left(t\right),y\left(t\right),{y}_{j}\left(t\right)\right)|\left(x\left(t\right),y\left(t\right),{y}_{j}\left(t\right)\right)\in \left[0,\stackrel{¯}{L}\right]×\left[0,\stackrel{¯}{L}\right]×\left[0,\stackrel{¯}{L}\right]\right\}$

4. 主要结果

${R}_{\text{*}}=\frac{{n}_{\infty }{b}_{\infty }{\text{e}}^{-{d}_{j}^{\infty }\tau }{\lambda }_{\infty }}{{d}^{\infty }\left({\mu }^{\infty }+{K}_{1}^{\infty }{\lambda }_{\infty }\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{R}^{*}=\frac{{n}^{\infty }{b}^{\infty }{\text{e}}^{-{d}_{j}{}_{{}_{\infty }}\tau }{\lambda }^{\infty }}{{d}_{\infty }\left({\mu }_{\infty }+{K}_{1}{}_{{}_{\infty }}{\lambda }^{\infty }\right)}$

4.1. 当 ${R}_{\ast }>1$ 时捕食者的一致持久性

${x}^{\prime }\left(t\right)=ax\left(t-\tau \right)-bx\left(t\right)-c{x}^{2}\left(t\right).$

1、如果 $a>b$ ，则 $\underset{t\to +\infty }{\mathrm{lim}}x\left(t\right)=\frac{a-b}{c}$

2、如果 $a ，则 $\underset{t\to +\infty }{\mathrm{lim}}x\left(t\right)=0$

, (11)

(12)

(13)

(14)

(15)

(16)

(17)

，从而，即，矛盾，从而(15)的结论是正确的。

1) 存在，当时，使得

2)附近振荡，并对所有T成立。

4.2.时捕食者的灭绝性

，以及文献 [25] 中的引理，可得，应用比较原理，最终可得当时，，显然也有，从而证明了成年捕食者的灭绝性。

5. 数值模拟

Figure 1. Basic behavior of solutions of system (3) with

Figure 2. The mature predator in system (3) is extinct

6. 结论

 [1] Fan, M. and Kuang, Y. (2004) Dynamics of a Non-Autonomous Predator-Prey System with the Beddington-DeAngelis Functional Response. Journal of Mathematical Analysis and Applications, 295, 15-39. https://doi.org/10.1016/j.jmaa.2004.02.038 [2] Hsu, S.B., Hwang, T.M. and Kuang, Y. (2003) Global Analysis of Michae-lis-Menten Type Ratio-Dependent Predator-Prey System. Journal of Mathematical Biology, 42, 489-506. https://doi.org/10.1007/s002850100079 [3] Jost, C. and Ellner, S. (2000) Testing for Predator Dependence in Predator-Prey Dynamics: A Nonparametric Approach. Proceedings of the Royal Society of London. Series B, 267, 1611-1620. https://doi.org/10.1098/rspb.2000.1186 [4] Rui, X., Davidson, F.A. and Chaplain, M.A.J. (2002) Persistence and Stability for a Two-Species Ratio-Dependent Predator-Prey System with Distributed Time Delay. Journal of Mathematical Analysis and Applications, 269, 256-277. https://doi.org/10.1016/S0022-247X(02)00020-3 [5] Wang, Y., Wu, H. and Sun, S. (2012) Persistence of Pollination Mutual-isms in Plant Pollinator-Robber. Theoretical Population Biology, 81, 243-250. https://doi.org/10.1016/j.tpb.2012.01.004 [6] Abrams, A. and Ginzburg, L.R. (2000) The Nature of Predation Prey Dependent, Ratio Dependent or Neither. Trends Ecology Evolution, 15, 337-341. https://doi.org/10.1016/S0169-5347(00)01908-X [7] Huang, C.-Y., Zhao, M. and Zhao, L.-C. (2010) Permanence of Periodic Predator-Prey System with Two Predators and Stage Structure for Prey. Nonlinear Analysis: Real World Applications, 11, 503-514. https://doi.org/10.1016/j.nonrwa.2009.01.001 [8] Beddington, J.R. (1975) Mutual Interference between Parasites or Predators and Its Effect on Searching Efficiency. Journal of Animal Ecology, 44, 331-340. https://doi.org/10.2307/3866 [9] Deangclis, D.L., Coldstein, R.A. and Neill, R.V. (1975) A Model for Trophic Interaction. Ecology, 56, 881-892. https://doi.org/10.2307/1936298 [10] Skalski, G.T. and Filliam, J.F. (2001) Functional Responses with Predator Inter-Ference: Viable Alternatives to the Holling Type Ii Model. Ecology, 82, 3083-3092. https://doi.org/10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2 [11] Cantrell, R.S. and Cosner, C. (2001) On the Dy-namics of Predator-Prey Model with the Beddington-DeAngelis Functional. Journal of Mathematical Analysis and Applications, 257, 206-222. https://doi.org/10.1006/jmaa.2000.7343 [12] Hwang, Z.W. (2003) Global Analysis of the Predator-Prey System with Beddington-DeAngelis Functional Response. Journal of Mathematical Analysis and Applications, 281, 395-401. https://doi.org/10.1016/S0022-247X(02)00395-5 [13] Liu, S. and Zhang, J. (2008) Coexistence and Stability of Predator-Prey Model of Bedding-DeAngelis Functional Response. Journal of Mathematical Analysis and Applications, 342, 446-460. https://doi.org/10.1016/j.jmaa.2007.12.038 [14] Aiello, W. and Freedman, H.I. (1990) A Time-Delay Model of Single-Species Growth with Stage Structure. Mathematical Biosciences, 101, 139-153. https://doi.org/10.1016/0025-5564(90)90019-U [15] Al-omari, J. and Gourley, S. (2003) Stability and Traveling Fronts in Lotka-Volterra Competition Models with Stage Structure. SIAM Journal on Applied Mathematics, 63, 2063-2086. https://doi.org/10.1137/S0036139902416500 [16] Liu, S., Chen, L., Luo, G. and Jiang, J. (2002) Asymptotic Behavior of Competitive Lotka-Volterrta System with Stage Structured. Journal of Mathematical Analysis and Applications, 271, 124-138. https://doi.org/10.1016/S0022-247X(02)00103-8 [17] Liu, S., Chen, L. and Agarwal, R. (2002) Recent Progress on Stage-Structured Population Dynamics. Mathematical and Computer Modelling, 36, 1319-1360. https://doi.org/10.1016/S0895-7177(02)00279-0 [18] Ou, L., Luo, Y., Jiang, Y. and Li, Y. (2003) The Asymptotic Behaviors of a Stage-Structured Autonomous Predator-Prey System with Time Delay. Journal of Mathematical Analysis and Applications, 283, 534-548. https://doi.org/10.1016/S0022-247X(03)00283-X [19] Liu, S. and Edoardo, B. (2010) Predator-Prey Model of Bedding-ton-DeAngelis Type with Maturation and Gestation Delays. Nonlinear Analysis, 11, 4072-4091. https://doi.org/10.1016/j.nonrwa.2010.03.013 [20] Liu, S. and Edoardo, B. (2006) A Stage-Structured Predator-Prey Model of Beddington-DeAngelis Type. SIAM Journal on Applied Mathematics, 4, 1001-1129. https://doi.org/10.1137/050630003 [21] Cushing, J.M. (1977) Periodic Time-Dependent Predator-Prey System. SIAM Journal on Applied Mathematics, 32, 82-95. https://doi.org/10.1137/0132006 [22] Yang, S.J. and Shi, B. (2008) Periodic Solution for a Three-Stage-Structured Predator-Prey System with Time Delay. Journal of Mathematical Analysis and Applications, 341, 287-294. https://doi.org/10.1016/j.jmaa.2007.10.025 [23] Zhang, T. and Teng, Z. (2007) On a Non-Autonomous Seirs Model in Epide-miology. Bulletin of Mathematical Biology, 69, 2537-2559. https://doi.org/10.1007/s11538-007-9231-z [24] Song, X. and Chen, L. (2001) Optimal Harvesting and Stability for a Two-Species Competitive System with Stage-Structure. Mathematical Biosciences, 170, 173-186. https://doi.org/10.1016/S0025-5564(00)00068-7 [25] Niu, X., Zhang, T. and Teng, Z. (2011) The Asymptotic Behavior of a Non-Autonomous Eco-Epidemic Model with Disease in the Prey. Applied Mathematical Modelling, 35, 457-470. https://doi.org/10.1016/j.apm.2010.07.010 [26] Gourley, S.A. and Kuang, Y. (2004) A Stage-Structured Predator-Prey Model and Its Dependence on Through-Stage Delay and Death Rate. Journal of Mathematical Biology, 49, 188-200. https://doi.org/10.1007/s00285-004-0278-2