Earth’s Human Carrying CapacityTerran WoolleyDifferential EquationsFinal ProjectTerran Woolley Earth’s Human Carrying CapacityIntroductionIn this presentation I will be exploring:the Human population growth rateWhat affects the growth rate?Is there a limit to the number of humans the earth cansupport? (carrying capacity)Logistic model for human population growthTerran Woolley Earth’s Human Carrying CapacityIntroductionIn this presentation I will be exploring:the Human population growth rateWhat affects the growth rate?Is there a limit to the number of humans the earth cansupport? (carrying capacity)Logistic model for human population growthTerran Woolley Earth’s Human Carrying CapacityIntroductionIn this presentation I will be exploring:the Human population growth rateWhat affects the growth rate?Is there a limit to the number of humans the earth cansupport? (carrying capacity)Logistic model for human population growthTerran Woolley Earth’s Human Carrying CapacityIntroductionIn this presentation I will be exploring:the Human population growth rateWhat affects the growth rate?Is there a limit to the number of humans the earth cansupport? (carrying capacity)Logistic model for human population growthTerran Woolley Earth’s Human Carrying CapacityUncertaintyThere have been many attempts to model the Earth’shuman population.Human population models are prone to errors.Will the population die off and leave a more reasonablenumber to support?At what point will the Earth’s resources be unable tosupport the human population?Terran Woolley Earth’s Human Carrying CapacityPast BehaviorThe annual rate of increase of the global population grew froma an average of 0.04% per year between A.D. 1 and 1650 to apeak of 2.1% around 1965 to 1970, then down to 1.6% per yearin 1995. The population rate has continued to decline and isnow at about 1.2%. It should be noted that world populationcalculations are prone to problems with accuracy.Terran Woolley Earth’s Human Carrying CapacityWhat affects the growth rate?Not taking into account for natural disasters that can affect thehuman population, the main factors that affect the growth rateof the human population are:food supplywater supplyTerran Woolley Earth’s Human Carrying CapacityWhat affects the growth rate?Not taking into account for natural disasters that can affect thehuman population, the main factors that affect the growth rateof the human population are:food supplywater supplyTerran Woolley Earth’s Human Carrying CapacityResourcesFoodPopulation that can be fed =food supplyindividual food requirementTerran Woolley Earth’s Human Carrying CapacityResourcesWaterPopulation that can be Watered =water supplyindividual water requirementTerran Woolley Earth’s Human Carrying CapacityResourcesFood and WaterPopulation that can be fed and watered= minimum ofnfood supplyindividual food requirement,water supplyindividual water requirementoTerran Woolley Earth’s Human Carrying CapacityThe Logistic EquationThe logistic equation is commonly used to model populationgrowth.P0= rP(1 − P/K )where P = P(t) is the population at time t, and K = K (t) is thecarrying capacity of the environment.Terran Woolley Earth’s Human Carrying CapacityThe Logistic EquationThe model will be in one of three states:If P(t) > K then P0(t) < 0 and the population will decrease.If P(t ) = K then P0(t) = 0 and the population will stay thesame.If P(t ) < K then P0(t) > 0 and the population will increase.Terran Woolley Earth’s Human Carrying CapacityThe Logistic EquationThe model will be in one of three states:If P(t) > K then P0(t) < 0 and the population will decrease.If P(t ) = K then P0(t) = 0 and the population will stay thesame.If P(t ) < K then P0(t) > 0 and the population will increase.Terran Woolley Earth’s Human Carrying CapacityThe Logistic EquationThe model will be in one of three states:If P(t) > K then P0(t) < 0 and the population will decrease.If P(t ) = K then P0(t) = 0 and the population will stay thesame.If P(t ) < K then P0(t) > 0 and the population will increase.Terran Woolley Earth’s Human Carrying CapacityThe Logistic EquationTerran Woolley Earth’s Human Carrying CapacityModifying the EquationThis model can be modified to fit the parameters of theparticular system. If we fit this model to the limiting resources,we can attempt to model the Human population.Terran Woolley Earth’s Human Carrying CapacityModifying the EquationFor example, Humans have the ability to create new ways ofgrowing and supplying food and water, therefore increasing theamount of people that can be fed and watered.Terran Woolley Earth’s Human Carrying CapacityModifying the EquationAn individual will, through his/her actions, cause the carryingcapacity to either:1Increase2DecreaseThis implies that Humans can affect their own carrying capacity.Terran Woolley Earth’s Human Carrying CapacityVariable carrying capacityTo incorporate this into our model we can let the constantcarrying capacity K in the logistic equation become a variableK (t).Terran Woolley Earth’s Human Carrying CapacityVariable carrying capacitySo the equation becomes:dP(t )dt= rP(t )[K (t ) − P(t)].Terran Woolley Earth’s Human Carrying CapacityVariable carrying capacityThe rate of change of the carrying capacity over time isproportional to the rate of change of the population over time.In other words:dK (t)dt= cdP(t )dtTerran Woolley Earth’s Human Carrying CapacityDefining cThe amount that an additional person can increase K (t)depends on the amount of resources available to maketheir hands productive.These resources are shared among more people as P(t )increases.Terran Woolley Earth’s Human Carrying CapacityDefining cThe amount that an additional person can increase K (t)depends on the amount of resources available to maketheir hands productive.These resources are shared among more people as P(t )increases.Terran Woolley Earth’s Human Carrying CapacityVariable cIf we replace the constant c for a variable c(t ) that decreasesas P(t ) increases. Letc(t) =LP(t )with L > 0Terran Woolley Earth’s Human Carrying CapacityCarrying CapacitySubstituting c(t) in for c we get:dK (t)dt=LP(t )dP(t )dt.Terran Woolley Earth’s Human Carrying Capacitygraph of a solution to P and KP(0) = 0.2523, K (0) = 0.252789, r = 0.0014829, L = 3.7Terran Woolley Earth’s Human Carrying CapacityProblems with the modelCannot accurately determine the value of L.Does not take into account for natural