PopulusHelp, biologia, Biologia I rok, od adama, studia, semestr I, Ekologia, ćwiczenia
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//-->ThePopulusHelp SystemWithPopulusrelease 5.3, we have changed the help system from a set of individual htmlfiles to a single large pdf document, accessible via Adobe Acrobat Reader. This willimprove the appearance of our help screens, and make future modifications much easier.To minimize downloaded file sizes, we will no longer provide both English and Spanishhelp files simultaneously; rather we will offer English or Spanish versions of the entirePopuluspackage. The Spanish version will show Spanish-language help files whenpossible, and English for those models that are not yet translatedPopuluswill start Acrobat Reader in the background, and display the full system whenhelp is requested, with a set of bookmarks for easy navigation. After looking at a helpscreen, we suggest that you minimize the reader rather than closing it, to speedsubsequent access. ThePopulushelp system is configured so that users may print pages(and we permit printing for any non-profit teaching use), but cannot extract, edit, or alterit for other applications.©D. N. Alstad, University of MinnesotaDensity-Independent Population GrowthDensity-independent growth models offer an extremely simple perspective on changes inpopulation size by assuming away many potential complications. For example, two sets ofcounteracting processes affect population size; birth and immigration increase populations whiledeath and emigration decrease them. To simplify, assume that (a) immigration and emigrationbalance, leaving birth and death as the only determinants of population density. Let's alsoassume that (b) all individuals are identical (especially with respect to their probabilities of dyingor producing offspring), (c) the population consists entirely of parthenogenetic females, so thatwe can ignore complications associated with mating, and (d) environmental resources areinfinite, so that the only factors affecting population size are the organisms' intrinsic birth anddeath rates. These assumptions allow a simplistic model of population growth, and it isinstructive to present the model in two formats for different kinds of life histories.Case I. Exponential Growth with Continuous BreedingFirst we will consider an organism likeHomo sapiensor the bacteria in a culture flask,with continuous breeding and overlapping generations. All ages will be present simultaneously,and population size will change steadily in small increments with the birth and death ofindividuals at any time. Thiscontinuouspopulation growth is best described by a differentialequation, with instantaneous rates defined over infinitely small time intervals.If:N= population sizeb= instantaneous birth rate per femaled= instantaneous death rate per femalethen population growth is given as:dN=(b−d)NdtIf we collect theper capitabirth and death rates in a single parameterr=b-dcalled theintrinsic rate of increaseorexponential growth rate,then:dN=rNdtThis expression states that population growth is proportional toNand the instantaneous growthrate,r.Whenr= 0, birth and death rates balance, individuals just manage to replace themselves,and population size remains constant. Whenr< 0, the population shrinks toward extinction, andwhenr> 0, it grows.We integrate the differential form of this continuous growth model to project futurepopulation sizes:N(t)=N()ert©D. N. Alstad, University of MinnesotaAlthoughris an instantaneous rate, its numerical value is only defined over a finite interval. Ifthis rate remains constant, then we can predict future population size,N(t)from a knowledge ofthe constant growth rate (r), the present population size,N(0),and the time over which growthoccurs (t).Case II. Geometric Growth with Discrete GenerationsNow we consider a density-independent growth model that is more appropriate for manyplants, insects, mammals, and other organisms that reproduce seasonally. Individuals in such apopulation comprise a series ofcohortswhose members are at the same developmental stage.Assume that an interval begins with the appearance of newborns, and that if individuals survivelong enough, they produce another cohort of offspring at the beginning of the next interval.Parents may all die before the offspring are born (like annual plants), or they may survive toreproduce again so that generations are partially overlapping (like many mammals). In eithercase youngsters appear in nearly synchronous groups separated by intervals without recruitment.Thisdiscretepopulation growth is best described by a finite difference equation.If:Nt= population size at time tb= births per female per intervalp= probability of surviving the interval, then:Nt+1=pNt+pbNt=(p+pb)NtRedefining the collective term with birth and death rates as a single parameterλ= (p +pb),which gives the number of survivors plus their progeny,Nt=λNt−1=λ(λNt−2)=λtNλis thegeometric growth factor,orper capitachange in population size over a discrete interval,t.Ifλ= 1, then individuals just manage to replace themselves and population size remainsconstant. Ifλ< 1, the population shrinks toward extinction, and ifλ> 1, it grows larger. As longasλremains constant, we can predict future population sizes from the growth rate (λ), the presentpopulation size (N), and the interval over which growth occurs (t), using the equationNt=λtNReferencesAlstad, D. N. 2001.Basic Populus Models of Ecology.Prentice Hall. Upper Saddle River, NJ.Chapter 1.Case, T. J. 2000.An Illustrated Guide to Theoretical Ecology.Oxford University Press. NewYork. pp. 1-13.Cohen, J. E. 1995.How Many People Can the Earth Support?W. W. Norton & Co. New York.©D. N. Alstad, University of MinnesotaElton, C. 1958. The Ecology of Invasions by Animals and Plants. Methuen, London.Ricklefs, R. E., and G. L. Miller. 1999.Ecology(4th edition). W. H. Freeman and Co., NewYork. pp. 298-302.Roughgarden, J. 1998.Primer of Ecological Theory.Prentice Hall, Upper Saddle River, N. J. pp.55-60.von Foerster, H., P. M. Mora and L. W. Amiot. 1960. Doomsday: Friday, 13 November, A.D.2026. Science 132:1291-5.© D. N. Alstad, University of MinnesotaDensity-Dependent Population GrowthThis module simulates density-dependent population growth, assuming a linear negativefeedback of population size onper capitagrowth. It requires specification of a starting populationsizeN(0),a maximum sustainable population size or environmental carrying capacityK,apercapitaintrinsic growth rater,and (optionally) a feedback lagτ.The program includes continuous,lagged continuous, and discrete simulations.Density-dependent models assume that population size affectsper capitagrowth. While thefeedback of density on growth can take many forms, the logistic model imposes a negative linearfeedback. Note that ifKis the environmental carrying capacity (quantified in terms of individuals,N),thenK–Ngives a measure of the unused carrying capacity, and (K -N)/Kgives the fraction ofcarrying capacity still remaining. ThusdNK−N=rNdtKFHIKIfNis near zero, the carrying capacity is largely unused, and dN/Ndt is nearr.IfN=K,theenvironment is totally used or occupied, and dN/Ndt = 0. In this continuous, differential equationmodel,ris an instantaneous rate, but its numerical value is defined over a finite time period.To project a time trajectory of logistic population growth, we need to integrate thedifferential equation from time (0) to time (t).N(t)=KK−N()−(rt)e1+N()A plot ofN(t)with respect to time gives a sigmoid (S-shaped) trajectory, where growth is nearlyexponential whenNis near zero, and slows to equilibrium atN=K.When initial population sizeexceeds the carrying capacity, numbers fall in an asymptotic approach towardKSometimes the feedback of density on per capita growth rate is not instantaneous. Forexample, the effect of malnutrition on population growth might not be strongly evident beforemalnourished juveniles reach reproductive age. We can simulate this process by assuming thatgrowth rates are affected by population size in some previous time period. ThusN(t−τ)dN=rN(t)1−Kdtwhereτis a time lag. There is no definite integral for this equation, so we project time trajectoriesby summing instantaneous changes in population size via numerical integration. Because the lag isdelayed by an amountτ,a growing population may reach and overshoot the carrying capacitybefore the negative feedback term causes the population to stop growing or decline. The resultingoscillation may damp to a stable equilibrium or continue indefinitely as a limit cycle.
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