Biological Models with Time Delay Differential Equations
Essay by Arindam Paul • October 31, 2017 • Study Guide • 1,218 Words (5 Pages) • 962 Views
Formulation equation, Equilibrium point and positivity test of model
The corresponding model is given by the following equations:
[pic 1]
[pic 2]
[pic 3]
Parameter | Descriptions |
o | Concentration of oxygen at time t. |
p | Density of phytoplankton. |
z | Density of zooplankton. |
A | Environmental factor on the rate of Oxygen production. |
f(o) | Concentration of dissolve oxygen function. |
g(o,p) | Function of phytoplankton growth rate. |
[pic 4] | Oxygen Consumption by Phytoplankton. |
[pic 5] | Oxygen Consumption by Zooplankton. |
m | The co-efficient of rate oxygen loss due to natural depletion. |
e(p,z) | Feeding of Zooplankton on Phytoplankton. |
k(o) | Consumed phytoplankton biomass is transformed into zooplankton biomass with efficiency. |
[pic 6] | Natural mortality rates of phytoplankton. |
[pic 7] | Mortality rates of phytoplankton by disease. |
[pic 8] | Natural mortality rates of zooplankton. |
monotonously decreasing function of o that tends to zero when the oxygen concentration in the water is becoming very large i.e . The above features are qualitatively taken into account by the following parametrization:[pic 9][pic 10]
[pic 11]
is the half-saturation constant.[pic 12]
The parametrization of plankton respiration see the second and third term in the right hand of the equation (1)
[pic 13]
is the maximum per capita phytoplankton respiration rate and [pic 15] is the half-saturation constant.[pic 14]
Regarding the zooplankton respiration, for many zooplankton species their oxygen consumption is known to depend on the oxygen concentration. The simplest parametrization of this kinetics is the Monod function:
[pic 16]
is the maximum per capita zooplankton respiration rate and [pic 18]is the half-saturation constant.[pic 17]
Considering phytoplankton multiplication, we assume that [pic 19]
where the first term describes the phytoplankton linear growth and second term accounts for intraspecific competition. Here [pic 20] per capita growth rate and [pic 21] describe the intraspecific competition.
The simplest parametrization for [pic 22]is
[pic 23]
is maximum phytoplankton per capita growth rate and [pic 25] is the half-saturation constant.[pic 24]
Thus for [pic 26], we obtain:
[pic 27]
Now for a prey-predator system with the logistic growth for prey (phytoplankton):
[pic 28]
where the carrying capacity:
[pic 29]
The logistic growth for predator (zooplankton):
[pic 30]
For predator use the following standard parametrization for predation:
[pic 31]
[pic 32] is the maximum predation rate and [pic 33] is the half-saturation prey density.
Finally, with regard to the zooplankton feeding efficiency as a function of the oxygen concentration, [pic 34]thus
[pic 35]
where [pic 36] is maximum feeding efficiency and [pic 37] is the half-saturation constant.
Now equation model takes the following more specific form:
[pic 38]
[pic 39]
[pic 40]
Due to their biological meaning, all parameter are nonnegative.
Critical Point:
Now steady states or Equilibrium points the model is steady state if
[pic 41], so we can write
[pic 42]
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