GSJ: VOLUME 6, ISSUE 8, August GSJ: Volume 6, Issue 8, August 2018, Online: ISSN

Transcription

1 GSJ: VOLUME 6, ISSUE 8, August GSJ: Volume 6, Issue 8, August 2018, Online: APPLICATION OF MATHEMATICAL MODELING ON POPULATION CARRY CAPACITY A CASE STUDY OF SOKOTO SOURTH LOCAL GOVERNMENT. Isah Mu azu & Abduladir A. Ibrahim isahisa2@gmail.com Phone: Lecturer Deartment Mathematics, Sooto State University ABSTRACT Methods and ways of redicting the future oulation as well as carrying caacity of the Sooto Source Local Government are being considered and taen care u using Malthusian law of Poulation. A model is generated that will be used to estimate the future oulation for Sooto Source Local Government. It was found out that the oulation of the Sooto Sourth Local Government will attain equilibrium at P (2292) = 1, i.e. when t =282 in the year The leadershi of the Sooto Sourth Local Government can now understand their oulation growth better have been established for them a model that will hel rovide them with the needed information of their oulation growth to aid roer lanning and distribution resources. 1.0 INTRODUCTION Mathematical modeling is the art of translating roblems an alication area into tractable mathematical formulations whose theoretical and numerical analysis rovides insight, for the original alication. This Research focuses mainly on oulation models. Although the level of mathematics required for this research wor is not difficult, setting u and maniulating models requires through effort and usually discussion after roblem have many answers due to different models used can illuminate different fact of the roblem. The objective is to rovide an aroach to roblems using some Mathematical methodology. Two models-exonential growth model and logistics growth model are oular in research of the oulation growth. The exonential growth model was roosed by Malthus in 1798, and it is also called the Malthusian growth model. However, when it comes to human oulation, whether this constant growth rate can be observed is in doubt. This is mainly because we can hardly define the conditions. Human reroduce

2 GSJ: VOLUME 6, ISSUE 8, August sexually and have consciousness. We cannot be sure under what condition will an individual be reroduced. Therefore, it is in doubt whether models based on constant growth rate can exlain human oulation growth. Taing carrying caacity into regards, logistic growth model imroves the receding exonential growth model. However, whether it can describe human oulation growth is in disute. The first imortant exercise in every modeling endeavor is to choose the facts of the investigated system that we wish to cature in our model. As I mentioned earlier before, this research wor is restricted to Sooto South local government oulation, there are only two methods used in this research namely exonential growth model and logistic model. 1.1 EXPONENTIAL GROWTH; Exonential growth is exhibited when the growth rate of the function current value, resulting in its growth with time being an exonential in a function in which the same way when the growth rate is negative. In case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay, the function values forming a geometric rogression. In either exonential decay, the ratio of the rate of change of the quantity to its current size remains constant over time (Malthus T.R. 1798). 1.2 LOGISTIC MODEL Logistic Model is a tyical alication of the logistic equation is a common model of oulation growth originality due to Pierre Francios Verhulst in 1838 where the rate of reroduction is roortional to both the existing oulation and the amount of available resources, all else being equal (Verhulst P.F. 1838) AIMS AND OBJECTIVE OF THE STUDY. A mathematical model achieving the twin objectives of simlicity and accuracy for the simulation of immobilized aced bed fomenters. The main aim and objectives of this wor is i. to introduce a very simle model in order to interret data and mae demograhic rojection and now its carrying caacity. ii. to roduce the most common quantitative aroaches to oulation dynamics, emhasizing the different theoretical foundation and assumtion. iii. to redict future oulation of the study area, using the modeling aroach on the osition of oulation dynamics.

3 GSJ: VOLUME 6, ISSUE 8, August REVIEW OF RELATED LITERATURE Dingyadi M. A. in her roject (2012) titled the mathematical modeling on oulation model, a case study of Bodinga Local Government; use of Malthusian law of oulation model to find out the ercentage that the oulation of Bodinga Local Government is increasing. Using the census obtained in According to Malthus ( ), nature has built in systems of checs and balances that limit the amount of damage that can be caused by a large grou by reducing the oulation of that grou once it increase a certain oint. Similarly, Malthus ut forth an exonential growth model for human oulation and finally concluded that eventually, the oulation would exceed the caacity to grow and adequate food suly. Similar studies have been carried out more frequently at a national level. Ulrich (1998), however, argued that immigration can only slow an inevitable decline of the oulation of Germany. He alied different fertility assumtions for natives and foreigners and different immigration levels by grou of immigrants, and estimated the oulation size of Germany and its structure in Wanner (2000), Elbar. L.L and Bereiric J.M (2002) in their study also showed that in the absence of future migration the total oulation of the country would start declining much earlier and would be 5.6 million in 2050, about 1.5 million less than what is currently rojected. The Lota Voltera redator-rey equations are first-order and nonlinear differential equation defined as; dv = av bv, d = c + dv Where the classic model a,b,c,d are all ositive constants A Malthusian growth model, sometimes called a simle exonential growth model, is essentially exonential growth loosed on a constant rate. The model is named after Thomas Robert Malthus (1798), who wrote an essay on the rincial of oulation, Malthusian models have the following form P(t) = P0ert where P(t) = P0 is the initial oulation size, r= the oulation growth rate (Malthusian arameter), and t = time.

4 GSJ: VOLUME 6, ISSUE 8, August METHODOLOGY Let (1) be the oulation of Sooto South Local Government at anytime t. Now considering (t) being continuous there is need to determine the two ratios, namely, growth and death ratio. But in this case we assume the death rate to be zero and growth rate to be roortional to the oulation resent; hence this assumtion is consistent with observation of growth rovided. Thus, from the assumtion that growth rate is roortional to the oulation resent for time t. we have (Chase J.M et. Al 2002): d, Now, from equation (1) we have by searating variables d = d = (1) We integrate both sides d =, ln = t + c Tae the exonential of both sides = e t+c, = e t.e c, = me t But the oulation is at time t. hence (t) = me t...(2) Initially if we let t = 0, we have P(0) = me (0) P(t) = P 0 e t (3), Where m = P 0 Equation (3) is called the Malthusian law of oulation growth model and the so called equation redicts that oulation always grows exonentially with time. Now we have to aly the equation to find out whether the oulation of Sooto South Local Government is increasing or decreasing by solving some numbers of roblems, using the rojection of census of 2010 as comiled by National Poulation Commission Sooto South Local Government of Sooto State.

5 GSJ: VOLUME 6, ISSUE 8, August Problem 1. The oulation of Sooto South Local Government in the year 2010 and 2011 are and resectively. Using Malthusian model we can estimate that oulation of Sooto South Local Government as a function of time. Let t = 0 to be the year in 2010 and P 0 = then from equation (3) P(t) = e t.(4) Now, we observe that in the year 2011 t = 1 (i.e ), then equation (4) becomes; (P)(1) = e (1) = e = To solve for, we tae the natural in of both sides. Thus, = Now, equation (4) becomes, P(t) = e 0.032t (a) From the result obtained in 1 to 6 above, we observed that, the oulation is increasing in three closed ercentage which disclosed the increasing rate to be 3.2%. Thus, using the ercentage above, we can find the true increase in the oulation of Sooto South Local Government. We now, consider other roblems that gives us the reality of the ercentage that the oulation of Sooto South is increasing.

6 GSJ: VOLUME 6, ISSUE 8, August Problem 2. The oulation of Sooto South Local Government in 2010 and 2012 are and resectively. We can estimate the oulation as a function of time t. Let t = o be the year 2010 and P o = then from equation (3) (t) = e t.(v) We observe that in the year 2012 t = 2 i.e. ( ) Thus, P(2) = e (2) = e 2 = Tae thenatural In of both sides to find the value of In (224681e 2 )= In , In Ine 2 = In239530, 2 = In In = , K =0.032 Substituting into equation (5) we have; (t) = e 0.032t (b) Problem 3 In 2010 the oulation of Sooto South Local Government was found to be and in 2013 it was , we estimate the oulation of Sooto South Local Government as a function of time t using equation (3) Let t = o be the year 2010, then equation (3) becomes: (t) = e t.(vi) To determine we observe that in 2013 when t =3, we have, P(3) = e 3 = e 3 = , Tae the natural In of both sides, In (224681e 3 ) = In In Ine 3 = In247319, 3 = In In = , 3 = 0.096, K = (by dividing both sides by 3) Now, equation (6) becomes (t) = e 0.032t (c)

7 GSJ: VOLUME 6, ISSUE 8, August Problem 4 If in the year 2010 and 2014 the oulation of Sooto South was found to be and resectively, then we can estimate its oulation as function of time using Malthusian model. : Suose t = o is the year 2010, then we have; P(t) = e t But in 2014, t = 4 and P(t) = , so we can determine the value of (t) = e t.(vii) e 4 = tae the natural in of both sides, ln (224681e 4 ) = ln255361, ln lne 4 = ln e 4t 4 = ln ln224681, 4 = = 0.128, K = Now, equation (7) becomes; value of. (t) = e 0.032t.(d) If we substitute the Problem 5 In the year 2010 and 2015 the oulation of Sooto South Local Government was found to be and resectively. We estimate its oulation using Malthusian model as a function of t. We let t = o in the year To determine the value of from equation (3) we have t = 5 (i.e ) in year 2015 (t) = e t.(viii) P(5) = e 5 = , Where P(t) = e 5 = , tae the natural ln of both sides, ln(224681e 5 = tae the natural ln of both sides, ln(224681e 5 ) = ln263665, ln lne 5 = ln263665, = ln ln = , K = Substituting into equation (8) we have, (t) = e o.o32t (e)

8 GSJ: VOLUME 6, ISSUE 8, August Problem 6 In 2010 the oulation of Sooto South was and in Then we estimate the oulation of Sooto South as a function of time using same model. If t = 0 be the yeat 2010, the by equation (3), (t) = e t..(9), But in the year 2016 t = 6 P(6) = e 6 = , e 6 = Tae the natural ln of both sides, ln(224681e 6 ) = ln ln lne 6 = ln272239, 6 = ln ln = , K = Put in equation (9) As a result of the above solved roblem from 1 to 6, we found out that oulation is increasing in three closed ercentage which show that the estimation was 3.2%. Thus, using the ercentage above one can find the truth increase of Sooto South Local Government oulation. We now, consider other roblems that that gives us the reality of the ercentage that the oulation of Sooto South is increasing. If the oulation of Sooto South Local Government was found to be in 2010 and increasing at rate of 3.2% er year when the oulation reach

9 GSJ: VOLUME 6, ISSUE 8, August Problem 7 d(t) =, by searating the variable we have, d(t) = but = 3.2% = 0.032, d(t) = 0.032, by integrating both sides we have; ln = 0.032t + c Tae the exonential of both sides P = e c, P = P 0 e 0.032t where P 0 = e c We are measuring time from 2010 when P 0 = , then equation (2) gives P(t) = e 0.032t So we calculate t when P(t) = , e 0.032t = , 0.032t = 0.032, t = 1 Problem 8 t = 1 year i.e. it reaches in ( ) = 2007 If the oulation of Sooto South was found to be in 2010 and is increasing at the rate of 3.2% annually, when will the oulation reaches ? We now that = 0.032, d(t) = 0.032, Searate the variable, d(t) = Integrate both sides ln = 0.032t + c Tae the exonential of both sides P(t) = e 0.032t+c, P(t) = P 0 e 0.032t We calculate t when P(t) = and P 0 = e 0.032t = Tae the natural ln of both sides ln t = ln239530, 0.032t = t = 2 years, i.e = 2012

10 GSJ: VOLUME 6, ISSUE 8, August Problem 9 If the oulation of Sooto South was found to be in 2010 and increases at a rate of 3.2% er year then the oulation will reached when: d(t) d(t) = where = = 0.032, (by searating variable) By integrating both sides we have; ln P(t) = 0.032t + c tae the exonential of both sides, P(t) = e 0.032t+c P(t) = P 0 e 0.032t, where P 0 = e c, Use P(t) = and P 0 = to find t e = Tae the natural ln of both sides ln(224681e 0.032t ) = ln t = 0.096, t = 3years Hence, the oulation will reach in ( ) = 2013 Problem 10 If the oulation of Sooto South was in 2010 and was found to be increasing at 3.2% annually, when will the oulation reaches ? d(t) =, where = d(t) = 0.032, d(t) = 0.032, (by searating variables) Integrate both sides, we have, ln(t) = 0.032t + c Tae the exonential of both sides, P(t) = e 0.032t+c, P(t) = P 0 e 0.032t, Where P 0 = e c We are measuring time from 2010 when P 0 = and calculating time when P(t) = e 0.032t = , Tae the natural ln of both sides, ln(224681e 0.032t ) = ln t = 4 years (by dividing both sides by 0.032). Hence the oulation will reach in ( ) = 2014 Problem 11

11 GSJ: VOLUME 6, ISSUE 8, August In 2010 the oulation of Sooto South was and is increasing at the rate of 3.2% er year, when will the oulation reaches ? d(t) = = d(t) = 0.032, (by searating variables) Integrate both sides, we have ln(t) = 0.032t + c tae the exonential of both sides P(t) = e0.032t+c, P(t) = P0 e0.032t but in 2010 where P(t) = and P0 = e0.032t= Tae the natural ln of both sides, ln(224681e0.032t) = ln263665, 0.032t = 0.16 t = 5 years (if is divided to both sides). Thus, the oulation will reaches in 2015 (i.e ) Problem 12 If the oulation of Sooto South was and increases at a rate of 3.2% er year, when will the oulation reach ? d(t) = = 0.032, d(t) = (by searating variables) Integrate both sides, ln(t) = 0.032t + c Tae the exonential of both sides P(t) = e0.032t+c, P(t) = P0 e0.032t But we are measuring time from 2010 when P0 = then we have P(t) = e0.032t So we calculate for t whenp(t) = Thus, e0.032t= , tae the natural ln of both sides ln(224681e0.032t) = ln272239, ln t = ln t = Divide both sides by 0.032, t = 6 years Therefore, the oulation reach in 2016 (i.e ). Moreover, for continuously increasing oulation without bound at t, the oulation reaches a oint where the environment can no longer suort it. We call this oint nown as the carry caacity of the environment (i.e. the number of eole that the environment can suort). If r is the growth constant then a reasonable modification of r to suort is given as

12 GSJ: VOLUME 6, ISSUE 8, August r = r, Recall that d = r..(i) Substituting for r in equation (i) we have, d = r (1 ), d = r(1 ), Which is the logistic equation. This form is a nonlinear ordinary differential equation of the first order. Now, we solve the differential equation, d Searating the variables we have, d = r (;) = r (;) d Multilying through by 1/( ), = r ; Integrate both sides, d/( ) We exress 1 d ; by artial fraction. Thus, d = A + B/( )..(ii) ; Multilying through by L.C.M. we have, 1 = A + B, 1 = A A + B A = 1, A = 1, Also, B = 1/ Substituting A and B into (ii) we have 1 ln ln( ;) + 1 ( ) = ( ) = r 1 ln + 1 ln = 1 ln + 1 = rt + c..(iii) ln ( ) = rt + c Solving for c at t = 0 and = 0 we have, ln P 0 ln( 0 )/ = c Substituting for c into equation (iii) we have ln ln ( ) = rt/ + lnp 0 / ln(p 0 )/ Multily through by we have, ln ln ( ) = rt + lnp 0 ln(p 0 ) By law of ln, ln ; Tae the exonential of both sides = rt + ln(p 0/(P 0 )) ; = ert. P 0 /(P 0 ).(iv) Cross multily P(P 0 ) = PP 0 e rt P 0 e rt, P(P 0 )PP 0 e rt = P 0 e rt Dividing through by P 0 e rt, P(P 0 )/P 0 e rt +P =.(v) P = P = P 0 P 0 e rt: P 0 e rt :1 ;e rt : P 0 e rt :1 P = ;e rt : e rt P 0 :1 P = ;1: P 0 e rt :1

13 GSJ: VOLUME 6, ISSUE 8, August P= ;1 P0 m (vi) e rt :1 In the limit, P(t) and the exression for P(t) gives the initial condition P = P0. Thus, this model redicts that the oulation increases steadily from the initial value to the oulation P0 and aroaches to carry caacity as time also increases, and following the assumtion made earlier that the oulation change d increases ositively. Without loss of generating let us state that to redict future oulation of Sooto South Local Government which is the case study of this roject wor, we first of all estimate the carrying caacity of the local government following its oulation dynamics. Thus an exression of equation for can be obtained from equation (iv) (Barned et.al 2009) i.e. (;) = P0 ert/(p0 ), Cross multily P (P0 ) = (P0 ert )(P ) We can get the arameters of the model from the oulation data (census of the local government of Sooto south. 3.2 PRESENTATION OF DATA The oulation of the eole of Sooto South Local Government Area ( ) The analysis here was based on the oulation of Sooto South Local Government which gives the estimate oulation of Sooto South from going by equation (iii) i.e. P(t) = P0 ert where, P0 is reresenting the initial oulation, P(t) is the resent or total oulation size K is growth rate constant and t is the time interval between the eriod of the oulation growth without loss of generality we can state that: P0 = and P(t) = if t is the initial time, then at 6 years interval = 6 Then is to be determined. P(t) = P0 et e6= = e6= = 6 = ln = From the revious roblems i.e. (roblem 1-6) and this analysis it imlies that r = is the growth rate er year, hence the ercentage rate of growth in Sooto South Local Government is = 3.2%. Based on the census of the oulation of Sooto South Local Government, the maximum number of individuals in which the local government can carry will be calculated from equation (vii).hence, K= PP0 ert ;PP0 P0 ert ;P Where P(t)= = P, P0 = , r = and t = 6. K = x224681e0.032x x

14 GSJ: VOLUME 6, ISSUE 8, August e 0.032x => = This imlies that, the realistic resources are exhausted when the oulation attains the equilibrium value i.e. when P(t) = and the oulation become more than the local government can carry resulting to cometition for sace, subsequent land disute, disease e.g. eidemic and finally over use of basic infrastructure. ANALYSIS OF THE FUTURE POPULATION OF SOKOTO SOUTH LOCAL GOVERNMENT AREA As we have seen how the oulation is increasing by the ercentage er year in roblem 7-12, we can now mae the analysis for future oulation of Sooto South Local Government and mae assumtions about redicted oulation. We can now redict and roject the oulation of Sooto South Local Government and relate it with the carrying caacity of the local government. Comaring with the environment, we have the required equation to forecast the future oulation using equation (vi) as follows: P (t) = ( 0 ;1) e rt :1 Now, we test the reality of this model by investigating whether the required oulation in 2016 will corresond with the one comiled by National Poulation Commission (NPC). Thus, P 0 = P = , r = 0.032, = , t = 6 P(2016) = e x P (2016) = Hence, the model is reliable to redict the future oulation of Sooto South Local Government from 2019 to By alying equation (vi) P(t) = 1 e P ;rt + 1 0

15 GSJ: VOLUME 6, ISSUE 8, August ANALYSIS OF WHEN THE POPULATION OF SOKOTO SOUTH LOCAL GOVERNMENT REACH IT CARRYING CAPACITY. Now, let analyze when the oulation will reach its carrying caacity i.e. using equation (iii) as follows: P(t) = P 0 e rt When P(t) = P 0 = and r = 0.032, then we can find the value of t = e 0.032t ln t = ln t = ln ln t = t = t = = 282, t = 282 years Hence, the oulation (i.e. carrying caacity ) reach in ( ) = This shows that in the year 2292 Sooto South Local Government reach a oulation that is more than it can carry, which result to cometition for saces and land disute and many more. The table below summarized the estimated oulation of Sooto South Local Government from 2019 to Table 4.1 Year Total Poulation , , , , , , , , , , , , , , , , ,963 Predicted Poulation of Sooto South Local Government.

16 GSJ: VOLUME 6, ISSUE 8, August SUMMARY In this research we considered methods and way of redicting the future oulation as well as the carrying caacity of the Study area using the formula of Malthusian law of oulation. In the research also some derivations of the formulae was achieved. The future oulation of the Sooto South Local Government in the next eighteen years (18yrs) was accomlished, this will really hel the Leadershi of the Local Government in lanning and the decision maing, thereby roviding clear icture of the exected oulation size for the next 18 years and hence the needed security, Healthcare, infra-structure and human caital develoment will be adequately taen care u for the wellbeing of the eole under the study area. CONCLUSION Based on the results obtained from the data analysis National Poulation Commission for Sooto South Local Government Sooto State, it was found that the Sooto South Local Government when it attains the equilibrium i.e. P(t)= (or in other word when it reaches the oulation i.e. the carry caacity of the Local Government), in essence the resources are exhausted and the oulation become larger than the local government can contain. And this will bring about cometition for sace, land disute, diseases out brea and many other survival challenges. This carrying caacity is = 1,858,432,481 (i.e. one billion, eight hundred and fifty eight million, four hundred and thirty two thousand, four hundred and eighty one). This oulation will be reached based on the rediction been made and formula used in the year when t=282 (i.e. in the year 2292). This shows when P (2292) = 1,858,432,481

17 GSJ: VOLUME 6, ISSUE 8, August REFERENCES Barnes, B. and Fulford, G R. (2009) Mathematical modeling with case studies: A differential Equations Aroach using male and Mattah 2 nd Editions. Chase, J.M., Abrams, P.A., Grover, J.P., Diehl, S., Chesson, P., Holt, R.D., Richards, S.A., Nisbet, R.M and case, J.J. (2002). The interaction between redation and cometitions: A review and synthesis. Ecology letters, 5, Dawed, M.Y., Koya, P.R and Goshu, A.T. (2014) Mathematical Modeling of Poulation Growth: The case of Logistic and Von Bertalanffy models. Oen Journal of Modeling and Simulation, 2, Dingyadi M.A (2012), Mathematical modeling on oulation models, A case study of Bondiga local government of Sooto State. Eberher, L.L. and Breiric, J.M (2012) Models for Poulation Growth Curves. ISRV Ecology, 2012, Article ID Goshu, A.T and Koya, P.R (2013) on of inflection oints of Nonlinear Regression Curvsimlications to statistics. American Journal of Theoretical and Statistics, 2, Gurevitech, J.J, Morrison, A. and Hedges, L.V (2000) The Interaction between Cometition and Predation: A Meta-Analysis of field Exeriments. The American Naturalist, 155, Koya, P.R and Goshu, A.T. (2013) Generalized Matematical Model for Biological growths. Oen journal of modeling and simulation, 1, Koya, P.R and Goshu, A.T (2013) solution of Rate-state Equation Decribing Biological Growths. American Journal of mathematics and statistics, 3, Malthus, T.R., (1798). Essay of the Princiles of Poulation.