Tuesday, February 2, 2010

Harvesting Model

Harvesting a Logistic Population:

When the harvesting term -k is incorporated into into bounded population model we have

[Graphics:Images/HarvestingModelMod_gr_11.gif].

There are three solution forms for this differential equation, and they correspond to the nature of the stationary solutions ( x(t) = c).

Definition(Stationary Points):

The stationary points of the D. E. [Graphics:Images/HarvestingModelMod_gr_12.gif] are solutions where [Graphics:Images/HarvestingModelMod_gr_13.gif] and are the roots of the characteristic equation

[Graphics:Images/HarvestingModelMod_gr_14.gif].

The roots are known to be [Graphics:Images/HarvestingModelMod_gr_15.gif], and the stationary solutions are [Graphics:Images/HarvestingModelMod_gr_16.gif].

Remark:

Since x(t) is a real function, there are no stationary solutions when [Graphics:Images/HarvestingModelMod_gr_17.gif].

Case (i) If [Graphics:Images/HarvestingModelMod_gr_18.gif] there is one stationary solution [Graphics:Images/HarvestingModelMod_gr_19.gif]:

When [Graphics:Images/HarvestingModelMod_gr_20.gif], the differential equation has the form [Graphics:Images/HarvestingModelMod_gr_21.gif] and the solution is

[Graphics:Images/HarvestingModelMod_gr_22.gif].
[Graphics:Images/HarvestingModelMod_gr_23.gif]

The solution with the initial condition [Graphics:Images/HarvestingModelMod_gr_24.gif] is

[Graphics:Images/HarvestingModelMod_gr_25.gif].

If [Graphics:Images/HarvestingModelMod_gr_26.gif] then [Graphics:Images/HarvestingModelMod_gr_27.gif].

If [Graphics:Images/HarvestingModelMod_gr_28.gif] then function x(t) has a vertical asymptote at [Graphics:Images/HarvestingModelMod_gr_29.gif]

and the population x(t) becomes extinct at some time [Graphics:Images/HarvestingModelMod_gr_30.gif] (where [Graphics:Images/HarvestingModelMod_gr_31.gif]), i. e.

[Graphics:Images/HarvestingModelMod_gr_32.gif].

Case (ii) If [Graphics:Images/HarvestingModelMod_gr_33.gif] there are two stationary solutions [Graphics:Images/HarvestingModelMod_gr_34.gif] and [Graphics:Images/HarvestingModelMod_gr_35.gif]:

When [Graphics:Images/HarvestingModelMod_gr_36.gif], the differential equation has the form [Graphics:Images/HarvestingModelMod_gr_37.gif] and the solution is

[Graphics:Images/HarvestingModelMod_gr_38.gif].
[Graphics:Images/HarvestingModelMod_gr_39.gif]

The two real roots of the characteristic equation [Graphics:Images/HarvestingModelMod_gr_40.gif], are [Graphics:Images/HarvestingModelMod_gr_41.gif].

The solution with the initial condition [Graphics:Images/HarvestingModelMod_gr_42.gif]



If [Graphics:Images/HarvestingModelMod_gr_44.gif] then [Graphics:Images/HarvestingModelMod_gr_45.gif].

If [Graphics:Images/HarvestingModelMod_gr_46.gif] then the population x(t) becomes extinct at some time [Graphics:Images/HarvestingModelMod_gr_47.gif], i. e. [Graphics:Images/HarvestingModelMod_gr_48.gif].

Case (iii) If [Graphics:Images/HarvestingModelMod_gr_49.gif] there are no stationary solutions:

When [Graphics:Images/HarvestingModelMod_gr_50.gif], the differential equation has the form [Graphics:Images/HarvestingModelMod_gr_51.gif] and the solution is

[Graphics:Images/HarvestingModelMod_gr_52.gif].
[Graphics:Images/HarvestingModelMod_gr_53.gif]

The solution with the initial condition [Graphics:Images/HarvestingModelMod_gr_54.gif] is



The function x(t) has a vertical asymptote at [Graphics:Images/HarvestingModelMod_gr_56.gif] so the population x(t) becomes extinct at some time [Graphics:Images/HarvestingModelMod_gr_57.gif] (where [Graphics:Images/HarvestingModelMod_gr_58.gif].), i.e.

[Graphics:Images/HarvestingModelMod_gr_59.gif].

No comments:

Post a Comment

Introductory Methods of Numerical Analysis