Question

3. A certain population of short lived insects is known to reproduce at a rate proportional to their current population. They are also known to reproduce at a rate proportional to the difference between the carrying capacity of their environment (the maximum population their envirnment can support) and their current population. The carrying capacity is proportional to their food supply, which changes with the season. The amount of food given in thousands of pounds is sin(t) + 2, where t is time measured in years. (a) Write down a differential equation for the population of these insects as a function of time. Indicate what each constant/symbol you use represents. (For example, P(t) = the total number of insects after t years.) (b) Describe what type of differential equation this is. (Use words such as 'order', 'linear', 'seperable', "reducible', 'homogenous', and 'autonomous ?) (c) Use Mathematica to solve this equation exactly.

Answer 1

t P(t) = population after years K= carrying capacity F(t)= food supply at time et Given- and dea (K-P(t)) Sin (t) + 2 dl & P(t) dt k & Flt) K= a (sint+2) {a= any constant} dP It & P (t) x (K- P(t)) & dp = bx (k P(t)- PIU) {b- any constant} dt dP dt = ab (sint+2) P(H) - b PIE (6) This is linear reducible diffeuential equations. ( because it can be reduced to linear foom).

(c)

write below equation in mathematica

In[1]:= DSolve[P'[t] == a*b*P[t]*(sin[t] + 2) - b*(P[t])^2, P[t], t ]

result will be -

Out[1] =

${{P[t] = \frac{e^{ab(-cost + 2t)}}{\int be^{ab(-cost + 2t)}dt}}}$

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