Consider the population x(t) of fish and y(t) of sharks at any time t in a certain region of the pacific ocean. Sharks prey on fishes and their population would decrease if isolated from the fish population. Let us assume further the following assumptions.
1) The change in the shark and fish population, in isolation, is respectively proportional to the present population of sharks and fish.
2) The number of sharks and fish caught by fisherman is directly proportional to the present population of the shark and fish population respectively. The fishing methods do not discriminate between sharks or fish.
3) The number of fish eaten by sharks is directly proportional to the product of the number of fish present and the number of sharks present.
4) The additional number of sharks surviving is directly proportional to the number of fish eaten. Under assumptions 1) to 4)
i) Formulate the mathematical model for the given problem and write a
system of differential equations governing it.
ii) Is the system of equations obtained in i) an autonomous system? Give reasons.
iii) Find the steady-state solution of the system in i) above. What does the solutions obtained correspond to?
iv) From the system of equations obtained in i) obtain the expression for
dy / dx and hence solve it under the condition x(0) = x0 , y(0) = y0 .
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A body of constant mass m is projected vertically upward with an initial velocity v0 in a medium offering a resistance k | v | , where k is a constant. Neglect changes in the gravitational force.
i) Find the maximum height xm attained by the body and the time tm at which this maximum height is reached.
ii) Show that if kv0 /mg < 1, then tm and xm can be expressed as
A particle is projected from the base of an inclined plane making an angle a with the horizontal. The initial velocity of projection is making an angle with the horizontal.
i) Derive the equation to the trajectory.
ii) Find the point at which the particle strikes the plane.
A particle moves from an initial velocity u at a constant acceleration f. Find an expression for the distance covered in time t using dimensional analysis.
Consider the following first-order ODE formulation = a{L- n(t)}, n(t0) = n0
.
Associate the variables {t, n(t)} and the parameters {a, L} to a physical system so that the above formulation becomes a mathematical model for population changes.
The rate of increase of susceptible SARS victims is proportional to the number of susceptible persons and number of infected persons. If there are S0 susceptible persons and 1-infected person at a time t0 then
i) set up the equation for the spread of the disease
ii) solve the resulting equation
iii) give a physical interpretation of the same by putting the epidemic curve
iv) write the limitations of this model.
A profit maximizing monopolist produces a quantity of a commodity. His
D(p) = 30 , where p is the price. Find the values of p, q and the profit.
Let a, K and 0 N be positive real numbers such that 0 < N0 <K. Find the solution of the logistic equation
If N0, N1, and N2 be the values of N at t = 0, t = T and t = 2T respectively, then show that
Also if at time t = t , the population N reaches half its carrying capacity then what is the form of the solution of the logistic equation?