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 N_1 + u(x,t)), N₂(x,t) = (\overline N_2 + v(x,t)), where |u|, |v| are small. Substituting into the system and retaining only linear terms gives ___ ______ ________ ____ ____ __________ __________.
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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?
The population dynamics of a species is governed by the discrete model
where r and k are positive constants. Determine the steady states and discuss the stability of the model. Find the value of r at which first bifurcation occurs. Describe qualitatively the behaviours of the population for , where
. Since a species becomes extinct if
for any n > 1, show using iterations, that irrespective of the size of r > 1 the species could become extinct if the carrying capacity
.
Maximize , subject to the constraints
and
and are integers.
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
The population consisting of all married couples is collected. The data showing the age of 12 married couples is as follows:
| Husband’s age (years) | Wife’s age (years) | Husband’s age (years) | Wife’s age (years) |
| 32 | 27 | 51 | 50 |
| 25 | 30 | 48 | 46 |
| 36 | 34 | 37 | 36 |
| 72 | 65 | 50 | 42 |
| 37 | 37 | 51 | 46 |
| 36 | 38 | 36 | 35 |
A library wants to improve its service facilities in terms of the waiting time of its borrowers. The library has two counters at present and borrowers arrive according to Poisson distribution with arrival rate 2 every 10 minutes and service time follows exponential distribution with a mean of 15 minutes. The library has relaxed its membership rules and a substantial increase in the number of borrowers is expected. Find the number of additional counters to be provided if the arrival rate is expected to be twice the present value and the average waiting time of the borrower must be limited to half the present value.