Logistic Growth model with Periodic Harvesting
Finally we assume seasonal breeding for the elephant population and model the population growth under periodic harvesting. This model is now periodic since the time dependent term has a periodic function. In this model r, K and are nonnegative parameters and H > 0.
In this model H denotes the contribution of the sine function to the growth of the population under the logistic growth model with periodic harvesting. Parameter ω represents the wavelength of the sine function that determines periodic harvesting.
The Logistic model with periodic harvesting translates to a differential equation which is non-autonomous having a period of T and depending explicitly on time.
We can qualitatively analyse the model by finding the stability …show more content…
This shows the existence of a critical value Hc which is a birfucation point for the model under a Poincare map.
4.1.11. Graphical Analysis
Figure 4.4: Graphical presentation of the harvesting process under three values of the harvesting parameter H under a Poincare map, Agudze Gilbert (April, 2013).
For values of H < Hc, there exist two periodic solutions and for H = Hc there is only one periodic solution. For a value of H above Hc, i.e H > Hc there are no fixed points for the model. Thus a larger rate of harvesting will lead to smaller populations which in future would result in the collapse of the population.
Modified Models 46
4.1.12. Logistic model with Poaching
Under this model we will consider harvesting with an element of poaching dP dt = rP(1 −
P
K
) − (H1 + H2) (4.21) where H1 is the number of elephants being harvested.
H2 is the number of elephants that have been removed from the population by poaching. The equilibrium points for the model above are obtained when dP dt = 0 and these are; 1. P1 =
K+
q
K2−
4K(H1+H2) r 2
2. P1 =
K−
q
K2−
4K(H1+H2) r 2
To determine the maximum sustainable harvesting rate H1, we equate the expression under the square root sign to 0 and solve for
Finally we assume seasonal breeding for the elephant population and model the population growth under periodic harvesting. This model is now periodic since the time dependent term has a periodic function. In this model r, K and are nonnegative parameters and H > 0.
In this model H denotes the contribution of the sine function to the growth of the population under the logistic growth model with periodic harvesting. Parameter ω represents the wavelength of the sine function that determines periodic harvesting.
The Logistic model with periodic harvesting translates to a differential equation which is non-autonomous having a period of T and depending explicitly on time.
We can qualitatively analyse the model by finding the stability …show more content…
This shows the existence of a critical value Hc which is a birfucation point for the model under a Poincare map.
4.1.11. Graphical Analysis
Figure 4.4: Graphical presentation of the harvesting process under three values of the harvesting parameter H under a Poincare map, Agudze Gilbert (April, 2013).
For values of H < Hc, there exist two periodic solutions and for H = Hc there is only one periodic solution. For a value of H above Hc, i.e H > Hc there are no fixed points for the model. Thus a larger rate of harvesting will lead to smaller populations which in future would result in the collapse of the population.
Modified Models 46
4.1.12. Logistic model with Poaching
Under this model we will consider harvesting with an element of poaching dP dt = rP(1 −
P
K
) − (H1 + H2) (4.21) where H1 is the number of elephants being harvested.
H2 is the number of elephants that have been removed from the population by poaching. The equilibrium points for the model above are obtained when dP dt = 0 and these are; 1. P1 =
K+
q
K2−
4K(H1+H2) r 2
2. P1 =
K−
q
K2−
4K(H1+H2) r 2
To determine the maximum sustainable harvesting rate H1, we equate the expression under the square root sign to 0 and solve for