1 Now we want the GLOBAL MAX of the crop s yield function YCN for N O find crit

1 now we want the global max of the crop s yield

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1 Now , we want the GLOBAL MAX of the crop 's yield function YCN ) for N > O find crit # s : Pnitrogen level yyn ) = C quotient rule ) cannot be negative . ( HNYZ Y' ( N ) = I - NZ zyz same domain as Y Soho type 2 crit # s type 't ? O= Y' l N ) 0=1 - NZ Enzi 0=1 - N ' 0=4 - NKHN ) go YIN ) has one critical # NE I ¥=-# not relevant for nitrogen level
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Example 13.2. Assume that the apples in an orchard grow logistically and are harvested according to the following DTDS: x t +1 = 2 . 5 x t (1 - x t ) - hx t the time step t is measured in weeks the quantity x t represents the portion of the maximum possible crop size (which has been estimated to be 100 000 kg of apples) available on week t the constant parameter h > 0 represents the harvesting intensity (fraction of available apples harvested each week) If this DTDS has a stable equilibrium, then in the long-term, we can assume the portion of available apples each week will approach the equilibrium quantity x . If we maintain a constant harvesting intensity h throughout the weeks of harvest, then the expected long-term weekly yield of this crop will approach Y ( h ) = hx . (a) Find the fixed point(s) of this system and the corresponding long-term yield (assuming this DTDS actually has a stable equilibrium). 2 intervals of 00 , 1) I ( I , D) domain co 2 sign of Y' ( N ) 440.5 ) > O Y' (3) SO on interval to -0 behaviour INCREASING DECREASING ooo the maximum yield is attained When of Yon interval I I Nitrogen levels are at 1mg 1kg soil ooo YIN ) has a LOCAL MAX at N=1 It is also a GLOBAL MAX since Yield The maximum yield Value is thus increases for OENCI , then decreases for all N > I Yf 1) = ¥150.5 units updating function Count fkf 2.5×4 - x ) - hx Ik ( or how greedily we harvest the available apples ⑧iB ) In the long-term , our weekly yield will approach Ylhkhx 't for fixed points ( also Known as " equilibria " or " steady States " ) , we solve X = fk ) x - - 2.5×4 - x ) - hx 0=2.5×-2.5×2 - hx - x O=X( 2.5-2 .5x - h -1 ) O=X( 1.5 - h - 2.54 ¥0 orb 1.5 - h -2.5×-0 2.5×-1.5 - h X = 1.5 - h Is =Q6 -0.4k go this DTDS has 2 equilibria : Emote inorderforthistobea positive x*=O
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