In a species of animals a constant fraction of the population = 6.2 are born each breeding season and a constant fraction ẞ=4.5 die. Formulate a difference equation for the population and find out the number of individuals after fifteen seasons given that the initial number is 987. Find the closed form solution of the formulated difference equation. If the growth rate of the population is represented by r then interpret the solution obtained when i) r > 0 and ii) r < 0.
غزل की परिभाषा और इसकी बाहरी संरचना पर विस्तृत चर्चा ग़ज़ल उर्दू साहित्य की सबसे लोकप्रिय और प्रभावशाली काव्य विधाओं में से एक है। यह प्रेम, सौंदर्य, जीवन दर्शन, वेदना, रहस्यवाद और विभिन्न मानवीय संवेदनाओं की अभिव्यक्ति का एक सशक्त माध्यम है। ग़ज़ल मूल रूप से अरबी साहित्य से आई और फ़ारसी होते हुए उर्दू साहित्य का अभिन्न अंग बन गई। यह एक विशेष शिल्प और संरचना के नियमों के तहत लिखी जाती है, जो इसे अन्य काव्य विधाओं से अलग पहचान देती है। ग़ज़ल एक संकलन होती है स्वतंत्र लेकिन आपस में किसी भावनात्मक या वैचारिक तार से जुड़ी हुई अशआर (शेरों) का। प्रत्येक शेर अपने आप में संपूर्ण होता है और उसका अर्थ अन्य शेरों से स्वतंत्र होता है। यही कारण है कि ग़ज़ल को एक विशेष संरचना की आवश्यकता होती है, जो इसकी खूबसूरती और प्रभावशीलता को बनाए रखती है। ### ग़ज़ल की बाहरी संरचना ग़ज़ल की संरचना निम्नलिखित तत्वों पर आधारित होती है: 1. मतला (مطلع) – ग़ज़ल का पहला शेर जिसे मतला कहा जाता है। इसमें दोनों मिसरे (पंक्तियाँ) एक ही क़ाफ़िया और रदीफ़ पर समाप्त होते हैं। यह ग़ज़ल का प्रारंभिक और अनिवार्य तत्व _________ _______ _____ _________ ______ _________ ________ __________ ________ _____ ___ ______.
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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.
Consider the following data
| x | 2 | 3 | 4 | 5 | 6 |
| y | 8.3 | 16.5 | 30.2 | 65.2 | 125.6 |
The owner of a readymade garments store sells two types of shirts: Zee-shirts and Button-down shirts. He makes a profit of Rs. 5 and Rs. 10 per shirt on Zee-shirts and Button-down shirt, respectively. He has two tailors, A and B at his disposal to stitch the shirts. Tailors A and B can devote at the most 7 hours and 15 hours per day, respectively. Both these shirts are to be stitched by both the tailors. Tailors A and B spend 2 hours and 5 hours, respectively in stitching one Zee-shirts, and 4 hours and 3 hours, respectively in stitching a Button-down shirt. How many shirts of both types should be stitched in order to maximize daily profit?
a) Formulate and solve this problem as an LP problem.
b) If the optimal solution is not integer-valued, use Gomory technique to derive the optimal integer solution.
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?
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.
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 1 every 6 minutes and service time follows exponential distribution with a mean of 10 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.