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.
### i) Rewrite the Equation and Identify the Growth Rate We start with the given difference equation: \[ N_{k+1} = \frac{\lambda N_k}{1 + a N_k} \] We need to express this in the form: \[ N_{k+1} = N_k + R(N_k) N_k \] Let's manipulate the given equation: \[ N_{k+1} = \frac{\lambda N_k}{1 + a N_k} \] Subtract \(N_k\) from both sides: \[ N_{k+1} - N_k = \frac{\lambda N_k}{1 + a N_k} - N_k \] Factor out \(N_k\) on the right-hand side: \[ N_{k+1} - N_k = N_k \left( \frac{\lambda}{1 + a N_k} - 1 \right) \] Simplify the expression inside the parentheses: \[ \frac{\lambda}{1 + ___ _______ ________ ______ ____.
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Assume that the return distribution on the two securities X and Y be as given below:
| Market 1 | Market 2 | Market 3 | |
| Probability | 2.0 | 5.0 | 3.0 |
| Security X | − 20% | 18% | 40% |
| Security Y | −10% | 20% | 15% |
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
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?
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 behaviors of the population for. Since a species becomes extinct if xn ≤1 1 for any n> 1, show using iterations, that irrespective of the size of r> 1 the species could become extinct if the carrying capacity k
Do the stability analysis of the following competing species system of equations with diffusion and advection
where Vand V, are advection velocities in x direction of the two populations with densities N, and N, respectively. a, is the growth rate, b, is the predation rate, d, is the death rate, C, is the conversion rate. D, and D, are diffusion coefficients. The initial and boundary conditions are:
where are the equilibrium solutions of the given system of equations. Interpret the solution obtained and also write the limitations of the model.
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.
For a multi-channel queuing system with λ = 12/hours, u=5/hours, c=3, po, = 0.056, calculate
i) The average time a customer is in the system
ii) The average number of customers in the system
iii) Whether any time would be saved for customers if the three-channel system with the service rate of 5 per hour is replaced by a single-channel system with an average service rate of 15 per hour?