Interaction between harvesting intervention and birth perturbation in an age-structured model

Abstract

An age-structured fish model with birth and harvesting pulses is established, where birth pulses are responsible for increasing the amount of fish due to the constant multiple placement of juveniles, and harvesting pulses describe the decrease of fish due to fishing activities. The principal eigenvalue as a threshold value depending on the harvesting and birth intensity is firstly investigated by three different ways. The asymptotic behavior of the population is fully investigated and sufficient conditions for the species to be extinct or persist are given. Numerical simulations suggest that interaction between negative harvesting intervention and positive birth perturbation decides extinction and persistence of the species. It is also shown that perturbation timing plays an important role.

Figures (5)

• Figure 1: The solution curve of $(N_2,\Lambda)$ in \ref{['7c']}. The red line represents the solution curve of equation $\Lambda=\frac{B_{11}-A_0N_2}{B_{12}-B_{13}N_2}$, which is strictly decreasing with respect to $N_2$ in interval $(-\infty,\frac{b}{A_0e^{(c_2-c_1)\tau}})\bigcup(\frac{b}{A_0e^{(c_2-c_1)\tau}},\infty)$. Also, it goes through fixed points $P_1(0,\frac{1}{(1+\delta)(1-\theta)})$ and $P_2(\frac{b}{A_0},0)$. One easily checks that $N_2^*=\frac{b}{A_0e^{(c_2-c_1)\tau}}$ and $\Lambda^*=\frac{1}{(1+\delta)(1-\theta)e^{(c_2-c_1)\tau}}$ are two asymptotics. Similarly, the green line is the solution curve of equation $\Lambda=\frac{A_0+B_{21}N_2}{B_{22}+B_{23}N_2}$, which is strictly increasing with respect to $N_2$ in interval $(-\infty,\frac{A_0}{ae^{(c_2-c_1)\tau}})\bigcup(\frac{A_0}{ae^{(c_2-c_1)\tau}},\infty)$. It goes through the fixed points $P_3(0,\frac{1}{1-\theta})$ and $P_4(-\frac{A_0}{a},0)$, and $N_2^{**}=-\frac{A_0}{ae^{(c_2-c_1)\tau}}$, $\Lambda^{**}=\frac{1}{(1-\theta)e^{(c_2-c_1)\tau}}$ are two asymptotics. Graph (a) is the case of $(1+\delta)e^{(c_2-c_1)\tau}<1$, Graph (b) is the case of $(1+\delta)e^{(c_2-c_1)\tau}>1$ and Graph (c) is the case of $(1+\delta)e^{(c_2-c_1)\tau}=1$, respectively.
• Figure 2: Simulations with: $\delta=0.7$, $\theta=0.1$ and $\tau=2$. Birth pulses happen at every time $t=0,2,4,\dots$ in $u_1$ and harvesting pulses occur at every time $t=1,3,5,\dots$ in $u_1$ and $u_2$. Graphs (a) and (b) are the projection of $u_1$ and $u_2$ in $t$ plant, respectively, Graph (c) is the cross-section view of $u_2$ in $t-x$ plant. It shows that juveniles and adults will spatially spread.
• Figure 3: Simulations with: $\delta=0.7$, $\theta=0.5$ and $\tau=2$. Graph (a) shows that birth pulse happen at time $t=0,2,4,\dots$ and harvesting pulse occur at time $t=1,3,5,\dots$. Graph (b) describes only impulsive harvesting occurs. Species eventually vanish.
• Figure 4: Simulations with: $\delta=2.7$, $\theta=0.5$ and $\tau=2$. Graphs (a)-(c) imply that juveniles and adults stabilized to a positive steady state.
• Figure 5: Simulations for $\tau=4$ and other parameters are the same as in Fig. \ref{['tu2']}. At every time $t=0,4,8,\dots$, birth pulses happen in juveniles and at every time $t=2,6,10,\dots$, harvesting pulses appear in juveniles and adults. Species finally converge to a positive steady state.

Theorems & Definitions (5)

• Example 3.1
• Lemma 3.1
• Theorem 3.2
• Theorem 4.1
• Theorem 4.2