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412 harvesting of renewable natural resources there

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4.12 Harvesting of Renewable Natural Resources There are many renewable natural resources that humans desire to use. Examples are fishes in rivers and sea and trees from our forests. It is desirable that a policy be developed that will allow a maximal harvest of a renewable natural resource and yet not deplete that resource below a sustainable level. We introduce a mathematical model providing some insights into the management of renewable resources. Let P(t) denote the size of a population at time t, the model for exponential growth begins with the assumption that kP dt dP = for some k>0. In this model the relative or specific, growth rate defined by P dt dP / is assumed to be a constant. 109
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In many cases P dt dP / is not constant but a function of P, let P / dt dP = f(P) or ) ( P f P dt dP = Suppose an environment is capable of sustaining no more than a fixed number K of individuals in its population. The quantity is called the carrying capacity of the environment. Special cases: (i) f (P)=c 1 P +c 2 (ii) If f(0)=r and f(K)=0 then c 2 =r and c 1 = - k r , and so (i) takes the form f (P) = r-( k r )P. Simple Renewable natural resources model is ) ( P K r r P dt dP - = This equation can also be written as ) ( bP a P dt dP - = Example 4.15: Find the solution of the following harvesting model 4 ) 5 ( - - = P P dt dP P(o)=P o Solution: 4.15 The differential equation can be written as 110
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) 1 )( 4 ( ) 4 5 2 ( - - - = + - - = P P P P dt dP or dt P P dP - = - - ) 1 )( 4 ( or dt dP P P - = - - - 1 3 1 4 3 1 Integrating we get c t 1 P 4 P ln 3 1 + - = - - or t e c P P 3 1 1 4 - = - - Setting t=0 and P=P 0 we find c 1 =(P o -4)/(P o -1). Solving for P we get t e Po Po t e Po Po t P 3 ) 4 ( ) 1 ( 3 ) 4 ( ) 1 ( 4 ) ( - - - - - - - - = 4.13 Exercises Newton’s Law of Cooling/Warming 1. A thermometer reading 100 0 F is placed in a pan of oil maintained at 10 0 F. What is the temperature of the thermometer when t=20 sec, if its temperature is 60 0 F when t = 8 sec? 2. A thermometer is removed from a room where the air temperature is 60 0 F and is taken outside, where the temperature is 10 0 F. After 1 minute the thermometer reads 50 0 F. What is the reading of the 111
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thermometer at t=2 minutes? How long will it take for the thermometer to reach 20 0 F . 3. Water is heated to a boiling point temperature 120 0 C. It is then removed from the burner and kept in a room of 30 0 C temperature. Assuming that there is no change in the temperature of the room and the temperature of the hot water is 110 o C after 3 minutes. (a) Find the temperature of water after 6 minutes (b) Find the duration in which water will cool down to the room temperature? Population Growth and decay 4. A culture initially has P o number of bacteria. At t=1 hour, the number of bacteria is measured to be 2 3 P 0 . If the rate of growth is proportional to the number of bacteria P(t) present at time t, determine the time necessary for the number of bacteria to triple. 5. Solve the logistic differential equation: 0 N ) 0 ( N , 0 t , N ) k N 1 ( o r dt dN = - = 6. Insects in a tank increase at a rate proportional to the number present. If the number increases from 50,000 to 100,000 in one hour, how many insects are present at the end of two hours.
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