Do the stability analysis of the following competing species system of equations with diffusion and advection
where V1 and V2 are advection velocities in x direction of the two populations with densities N1 and N2 respectively. a1 is the growth rate, b1 is the predation rate, d1 is the death rate, C1 is the conversion rate. D1 and D2 are diffusion coefficients. The initial and boundary conditions are:
at
and
where are the equilibrium solutions of the given system of equations.
Interpret the solution obtained and also write the limitations of the model.
The spatially homogeneous equilibria are obtained by neglecting diffusion and advection terms and setting the time derivatives equal to zero. The system then reduces to a₁N₁ − b₁N₁N₂ = 0, −d₁N₂ + C₁N₁N₂ = 0. From these equations the equilibrium points are (0,0), (a₁/b₁,0) and the coexistence equilibrium (\overline N_1 = d_1/C_1,\quad \overline N_2 = a_1/b_1.) The boundary conditions impose that the spatial solution is constant and equal to the equilibrium values at x = 0 and x = L, so stability is examined by considering small perturbations about the coexistence equilibrium. Let N₁(x,t) = (\overline __________ _____ ___ ____ _________.
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A company manufacturing soft drinks is thinking of expanding its plant capacity so as to meet future demand. The monthly sale for the past 6 years are available. State, giving reasons, the type of modelling you will use to obtain good estimates for future demand so as to help the company make the right decisions. Also state four essentials and four non-essentials for the problem.
A tumour is developing from the organ of a human body with concentration with growth and decay control parameters 7.2 and 2.7 respectively. In how many days the size of the tumor will be twice?
Let G(t) be the amount of the glucose in the bloodstream of a patient at time t. Assume that the glucose is infused into the bloodstream at a constant rate of . At the same time, the glucose is converted and removed from the bloodstream at a rate proportional to the amount of the glucose present. If at
then
i) formulate the model.
ii) find g(t) at any time t.
iii) discuss the long term behavior of the model.
Consider a discrete model given by
Investigate the linear stability about the positive steady state N by setting . Show that nt satisfies the equation
Hence show that is a bifurcation value and that as
the steady state bifurcates to a periodic solution of period 6.
Do the stability analysis of the following model formulated to study the effect of toxicant on prey-predator population and interpret the solution.
Where all the variables and constants are same as defined in the system (32)-(35) except for the following
Companies considering the purchase of a computer must first assess their future needs in order to determine the proper equipment. A computer scientist collected data from seven similar company sites so that computer hardware requirements for inventory management could be developed. The data collected is as follows:
| Customer Orders (in thousands) | Add-delete items (in thousands) | CPU time (in hours) |
| 123.5 | 2.108 | 141.5 |
| 146.1 | 9.213 | 168.9 |
| 133.9 | 1.905 | 154.8 |
| 128.5 | 0.815 | 146.5 |
| 151.5 | 1.061 | 172.8 |
| 136.2 | 8.603 | 160.1 |
| 92.0 | 1.125 | 108.5 |
Maximize , subject to the constraints
and
and are integers.