Population Dynamics Definition Model A model is defined as a physical representation of any natural phenomena Example: 1. A miniature building model.

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1 Popultion Dynmics Definition Model A model is defined s physicl representtion of ny nturl phenomen Exmple: 1. A miniture building model. 2. A children cycle prk depicting the trffic signls 3. Disply of clothes on models in sho room nd so on. Mthemticl Models A mthemticl model is representtion of phenomen by mens of mthemticl equtions. If the phenomen is groth, the corresponding model is clled groth model. Here e re going to study the folloing 3 models. 1. Liner model 2. Exponentil model 3. Logistic model Liner model The generl form of liner model is y = +bx. Here both the vribles x nd y re of degree 1. In liner groth model, the dependent vrible is lys the totl dry eight hich is noted by nd the independent vrible is the time denoted by t. Hence the liner groth model is given by = +. To fit liner model of the form y=+bx to the given dt. Here nd b re the prmeters (or) constnts to be estimted. Let us consider (x 1,y 1 ),(x 2, y 2 ) (x n, y n ) be n pirs of observtions. By plotting these points on n ordinry grph sheet, e get collection of dots hich is clled sctter digrm. In liner model, these points lie close to stright line. Suppose y = +bx is liner model to be fitted to the given dt, the expected vlues of y corresponding to x 1, x 2 x n re given by (+bx 1 ), (+bx 2 ), (+bx n ). The corresponding observed vlues of y re y 1, y 2 y n. The difference beteen the observed vlue nd the expected vlue is clled residul. The Principles of lest squres sttes tht the constnts occurring in the curve of best fit should be chosen such tht the sum of the squres of the residuls must be minimum. Using this for liner model e get the folloing 2 simultneous equtions in nd b, given by Σy = n+bσx (1) Σxy = Σx+bΣx (2)

2 here n is the no. of observtions. Equtions 1 nd 2 re clled norml equtions. Given the vlues of x nd y, e cn find Σx, Σy, Σxy, Σx 2. Substituting in equtions (1) nd (2) e get to simultneous equtions in the constnts nd b solving hich e get the vlues of nd b. Note: If the liner eqution is =+ then the corresponding norml equtions become Σ = n + bσt (1) Σt = Σt + bσt (2) x 1 y 1 x 3 y 3 0 y * * x 2 y 2 +bx 1 t x 1 There re to types of groth models (liner) (i) = + (ith constnt term) (ii) = (ithout constnt term) The grphs of the bove models re given belo : =+ = (1) (2) t =0

3 stnds for the constnt term hich is the intercept mde by the line on the xis. When t=0, = ie gives the initil DMP( ie. Seed eight) ; b stnds for the slope of the line hich gives the groth rte. Problem The tble belo gives the DMP(kgs) of prticulr crop tken t different stges; fit liner groth model of the form =+, nd lso clculte the estimted vlue of. t (in dys) ; DMP : (kg/h) Exponentil model This model is of the form =e here nd b re constnts to be determined Groth rte = d = be dt Reltive groth rte = 1 d. = dt be e = b Here RGR = b hich is lso knon s intensive groth rte or Mlthusin prmeter. To find the prmeters nd b in the exponentil model first e convert it into liner form by suitble trnsformtion. No = e (1) Tking logrithm on both the sides e get log e = log e (e ) log e = log e + log e e log = log e + log e e log = log e + Y = A (2) Where Y = log nd A = log e Here eqution(2) is liner in the vribles Y nd t nd hence e cn find the constnts A nd b using the norml equtions. ΣY = na + bσt ΣtY = AΣt + bσt 2 After finding A by tking ntilogrithms e cn find the vlue of

4 Note The bove model is lso knon s semilog model. When the vlues of t nd re plotted on semilog grph sheet e ill get stright line. On the other hnd if e plot the points t nd on n ordinry grph sheet e ill get n exponentil curve. Problem: Fit n exponentil model to the folloing dt. t in dys W in mg per plnt Logistic model (or) Logistic curve follos: The eqution of this model is given by = 1 kt + ce (1) Where, c nd k re constnts. The bove model cn be reduced to liner form s ce kt = 1 = 1 + ce kt = ce 1 kt Tking logrithm to the bse e, log log e 1 = log c kt 1 = log c + log e Y = A + Bt (2) Where Y = log e 1 A = log e c kt B = - k No the eqution (1) is reduced to the liner form given by eqution (2) using this e cn determine the constnts A nd B from hich e cn get the vlue of the constnts c nd k.

5 Problem The mximum dry eight of groundnut is 48 gms. The folloing tble gives the dry mtter production of groundnut during vrious dys estimte the logistic groth model for the folloing dt. t in dys DMP in gm/plnt