Modeling and numerical simulation of dissolved oxygen and biochemical oxygen demand concentrations with Holling type III kinetic relationships

On Modelling Water Quality With Stochastic Differential Equations

“…Hence, E s is a stable equilibrium point. To verify that the model has a stable interior equilibrium E s , we take parameter values as in [6]:…”

“…We consider the influence of noise for the same parameter values k 1 = 0.15, k 2 = 0.5, q = 0.2, x 2 c = 9.5, α = 0.35, k = 3, where the system in equation (2.1) possesses a stable equilibrium E s = (x 1 s , x 2 s ) = (1.89, 7.52). Resulting solutions for initial conditions (x 1 (0), x 2 (0)) = (12,6) are presented in Figures 2 and 3 In Figure 3, we show a plot of the sample paths of the solution (x 1 (t), x 2 (t)) for noise intensities σ 1 = σ 2 = 0.1. Figure 3 also shows sample paths of x 1 (t) and x 2 (t) for large noise intensities σ 1 = σ 2 = 0.4.…”

“…To analyse the stability of this equilibrium point, we determine the characteristic equation of the associated Jacobian matrix of equation (2.1) given by Using the Routh–Hurwitz criterion [2], which guarantees that all the roots of the characteristic equation have negative real part if and only if and , and considering that the model parameters are positive, the coefficients of the characteristic equation of the Jacobian matrix are Hence, is a stable equilibrium point. To verify that the model has a stable interior equilibrium , we take parameter values as in [6]: . The resulting phase portrait is shown in Figure 1.

Figure 1 This figure shows the phase portrait of equation (2.1) for given parameter values.

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“…They simulate the effects and interactions between dissolved oxygen (DO) and biochemical oxygen demand (BOD), algae and others. Aspects of these models have been considered by different authors (see, for example, [3, 6, 12] and the references therein). Moreover, stochastic differential equation models for the evolution of DO and BOD were developed [1, 4, 8, 11, 13, 14].…”

Nonlinear dynamics in a discrete-time predator-prey system

“…Among the variety of models, the predator-prey models with different Holling type functional response are refined so as to better reflect the specific characteristics of the different populations or economical needs. A number of interesting works exhibiting quite rich and complicated dynamics have been performed by the theoretical and mathematical biologists in the last few decades considering Holling type functional responses ( [7][8][9][10][11][12][13]). The literatures mentioned above investigate the stabilities and the analyses of codimension-one bifurcation of discrete predator-prey models.…”