User:Yongtao-Li/NM.HW7.1.2

= Problem 7.1.2: Use Hermite-Simpson Algorithm to Numerically Integrate the Verhulst Equation=

Given: The Verhulst Model for Logistic Population Growth
Verhulst(1838) equation

The Verhulst Model, as it pertains to logistic population growth, describes the following system dynamics.
 * LHS: The time derivative $$ \displaystyle \dot{x}(t)= \frac{dx}{dt} $$, is the rate that the population changes with respect to time.
 * RHS: Is directly proportionate to the following:
 * $$ \displaystyle x(t):= $$ the population size at time $$ \displaystyle t$$.
 * $$ \displaystyle \left(1-\frac{x(t)}{x_{max}} \right):= $$ the remaining available resources.
 * $$ \displaystyle x_{max}:= $$ the carrying capacity which is maximum number of inhabitants for an isolated system.
 * $$ \displaystyle r:= $$ the constant of proportionality which serves as the rate constant. We are considering the case of population growth where $$ \displaystyle r>1 $$.

2. Using the initial condition $$ \displaystyle x_0 = 9 > \frac{1}{2}x_{max} $$

 * Obtain the numerical solution by solving the ODE shown in equation (7.2.1.1) for the following parameters.
 * Given the population carrying capacity $$ \displaystyle x_{max}=15 $$.
 * Given the population growth rate $$ \displaystyle r = 1.4 $$.

Solution: Numerical Integration
The following Matlab codes are used to solve Eq. 7.1.2.1

1)$$ x_0 = 3$$



2)$$ x_0 = 9$$



3)comparison between the above cases



Comment
Form the above two cases, we find that these two cases, with different initial condition, can converge to the maximum value exactly. It's interesting that it seems that both cases achieve the maximum at the same time. However, the increase rate is different for they have various initial start point.