Time-delayed coupled logistic capacity model in population dynamics

This study proposes a delay-coupled system based on the logistic equation that models the interaction of a population with its varying environment. The integro-diferential equations of the model are presented in terms of a distributed time-delayed coupled logistic-capacity equation. The model eliminates the need for a prior knowledge of the maximum saturation environmental carrying capacity value. Therefore the dynamics toward the final attractor in a distributed time-delayed coupled logistic-capacity model is studied. Exact results are presented, and analytical conclusions have been done in terms of the two parameters of the model.

Figure 1

Stability analysis for $\lambda \lessgtr 1$ and for several values of $t_D$ using Eq. (22) to show graphically possible intersection points [roots of Eq. (10) using $z=c$, i.e., with $\omega =0$].

Figure 2

Andronov-Landau-Hopf bifurcation analysis from Eq. (10) using $z=i\omega$ (with $c=0$) for several values of $\lambda$ and $t_D$. (a) Zero cross points from Eq. (25) in the interval $\omega t_D\in (0,2\pi )$ for $t_D={10}^{-1}$ and $\lambda (=0.3,0.5,5,10,100)$. (b) Zero cross points from Eq. (25) in the interval $\omega t_D\in (0,4\pi )$ for $\lambda =0.5$ and $t_D(=1,5,10)$.

Figure 3

Numerical solutions from Eqs. (41, 42, 43) as a function of time for two set of initial conditions $\left\{N\right(0),K(0\left)\right\}$, using the Laplace distributed time-delayed coupled Logistic-Capacity model. Two values of $\lambda (=0.05,0.5)$ and different values of $t_D$ have been used. Solutions correspond to the approximation $O\left(t_D\right)$, so these evolutions are valid only for a short elapse of time. This crucial time $t^{\prime }$ can be observed by looking at the behavior of $V\left(t\right)$. This time $t^{\prime }$ corresponds to the time when having arrived the function at its stationary state, $V\left(\infty \right)=0$, it starts to abandon its plateau; this means that after this elapse of time the approximation $O\left(t_D\right)$ used in Eq. (43) does not work any more. The asymptotic value $\overline{N}$ can be observed to be in agreement with the Areas's rule Eq. (39) before the approximation $O\left(t_D\right)$ starts to fail. (a) The temporal behavior of $N\left(t\right),K\left(t\right)$, and $V\left(t\right)\equiv \dot{K}\left(t\right)$ for $N\left(0\right)=0.1$ and $K\left(0\right)=0.4$, using $\lambda =0.5$ and $t_D={10}^{-3}$. (b) $N\left(t\right),K\left(t\right),V\left(t\right)$ for $N\left(0\right)=0.1$ and $K\left(0\right)=0.2$, for the same values of $\lambda ,t_D$. (c) Corresponds to $\lambda =0.05$ with initial conditions $N\left(0\right)=0.1$ and $K\left(0\right)=0.2$, for two values of the delay time: $t_D={10}^{-4}$ (circles) and $t_D=0.5$ (squares).

Figure 4

Numerical solutions of $N\left(t\right),K\left(t\right),V\left(t\right)$ from Eqs. (41, 42, 43) using $\lambda =1,t_D={10}^{-3}$. Here it is possible to see that $V\left(t\right)$ crosses the zero value without approaching any plateau-like behavior as its does in Fig. 3.