Do the stability analysis of the following model formulated to study the effect of toxicant on one competing species where the environment toxicant concentration is being taken to change w.r.t. time.
along with the initial conditions.
Here,
N1 (t)=Density of prey population
N2(t)=Density of predator population
Co(t) = Concentration of the toxicant in the individual of the prey population
P=Constant environmental toxicant concentration.
a,, a, are the predation rates, r,r, are the growth rates or birth rates, d, is the death rate due to C, m, is the depuration rate, Q, h, k, g are positive rate constants.
To perform a stability analysis of the system described by the given differential equations, we need to find the equilibrium points and analyze their stability. Let's break the model into its components and proceed with the steps. The system of differential equations is: \[ \frac{dN_1}{dt} = r_1 N_1 - \alpha_1 N_1 N_2 - d_1 C_0 N_1 \] \[ \frac{dN_2}{dt} = r_2 N_2 - \alpha_2 N_1 N_2 \] \[ \frac{dC_0}{dt} = k_1 P - g_1 C_0 - m_1 C_0 \] \[ \frac{dP}{dt} = Q - hP - k P N_1 + g C_0 N_2 \] The variables in the system _________ ______ _________ _______ ________ ________ __________ __________ _______.
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Consider the following data
| x | 2 | 3 | 4 | 5 | 6 |
| y | 8.3 | 16.5 | 30.2 | 65.2 | 125.6 |
Five securities have the following expected returns
A=20%, B=15%, C=25%, D=22%, E=18%. Calculate the expected returns for a portfolio consisting of all five securities under the following conditions
i) The portfolio weights are of equal percentage in each
ii) The portfolio weights are 32% in A and remaining are equally divided among other four securities.
Ships arrive at a port at the rate of one in every 4 hours with exponential distribution of inter- arrival times. The time a ship occupies a berth for unloading has exponential distribution with an average of 10 hours. If the average delay of ships waiting for berths is to be kept below 14 hours, how many berths should be provided at the port?
A company has three factories that supply to three markets. The transportation costs from each factory to each market are given in the table. Capacities of the factories and market requirements are shown. Find the minimum transportation cost.
| M1 | M2 | M3 | ai | |
| F1 | 2 | 1 | 3 | 20 |
| F2 | 1 | 2 | 3 | 30 |
| F3 | 2 | 1 | 2 | 10 |
| bj | 10 | 10 | 20 | 40/60 |
Let P=(w1,w2) be a portfolio of two securities. If variance of P is minimum then find the value of w₁and w₂ in the following situations.
i) ρ12 = −1
ii)σ1 = σ2
iii)ρ12 = − ,5.0 σ1 = 5.1 and σ2 = .5.2
A model for insect populations leads to the difference equations
where λ and a are positive constants.
i) Write the equation in the form , and hence identify the growth rate.
ii) Plot the graph of R(Nk) as a function of N2
iii) Express the intrinsic growth rater and the carrying capacity K, for this model, in terms of the parameters, a and λ.
iv) Find the steady-state solution of this model and analyse the solution.
ompanies located on the banks of a river, dumping their chemicals waste into river, causing high levels of pollution. Local authorities passed new legislation with very high fines if the pollution in the river exceeds certain specified concentration limits. State, giving reasons, the type of modeling you will use to find a policy for discharging the waste to ensure that the concentration level never exceeds the specified limits. Also state four essentials and two non-essentials for the problem.