Ecology 302 Lecture III. Exponential Growth (Gotelli, Chapter 1; Ricklefs, Chapter 11, pp )

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1 Eclgy 302 Lecture III. Expnential Grwth (Gtelli, Chapter 1; Ricklefs, Chapter 11, pp ) Apcalypse nw. The Santa Ana Watershed Prject Authrity pulls n punches in prtraying its missin in apcalyptic terms. The riginal hrsemen were Cnquest (Pestilence), War, Famine and Death. 1

2 Key Pints. Ppulatin grwth (r decline) cnsequent t reprductin, death, immigratin, emigratin. Use difference equatins when generatins nn-verlapping; differential equatins when generatins verlap. Absent envirnmental checks, a clsed ppulatin grws withut limit r declines t zer. Expnential grwth: Gemetric grwth: In bth cases, the sle equilibrium is 0. If 0 is stable, the ppulatin cllapses; if 0 is unstable, the ppulatin expldes. The watershed between grwth and cllapse is Expnential grwth: 0; Gemetric grwth: 2 1.

3 Equilibria and stability. is an equilibrium f / if 0. is an equilibrium f if. An equilibrium is stable if, fllwing a small perturbatin, the system returns t equilibrium. An equilibrium is unstable if, fllwing a small perturbatin, the system mves further away frm equilibrium. Equilibrium des nt imply stability. Sme equilibria are stable; thers nt. Fr an expnentially grwing ppulatin, cnstant rates f immigratin and emigratin can shift the equilibrium density,, away frm zer, but d nt change its stability. Cnstant immigratin can maintain a cllapsing ppulatin. Cnstant emigratin cannt stabilize a grwing ppulatin. 3

4 Real wrld ppulatins are subject t stchastic influences. There are tw surces f randmness: Extrinsic (envirnmental fluctuatins); Intrinsic (prbabilistic nature f births and deaths). Latter prevails in small ppulatins. Imprtant cnsequences f randmness. Deterministic slutins are replaced by prbability distributins. Chance extinctin. The wrld s human ppulatin cntinues t grw, but at a less than expnential rate. Uncertainty regarding future human ppulatin size reflects the difficulty f predicting future rates f fertility and mrtality. 4

5 Demgraphic Transitin (DT). Histrically, ecnmic develpment accmpanied by reductins in mrtality (this cmes 1 st ) and fertility (this cmes 2 nd ). DT accmpanied by changing ppulatin age structure (increasing numbers f lder individuals). Increased rati f ecnmically prductive t dependent age classes cntributed t the Asian miracle. In many 1 st wrld cuntries, fertility has nw drpped belw the replacement rate 2 nd demgraphic transitin (DT2). As birth rates cntinue t decline the rati f ecnmically prductive t dependent age classes is again declining. 5

6 I. Expnential (Gemetric) Grwth. A. Ppulatin increase (r decrease) determined by births (B), deaths (D), immigratin (I) and emigratin (E). 1. Fr species with verlapping generatins, use differential equatins t mdel grwth e.g., / (1a) 2. Fr species with nn-verlapping generatins, use difference equatins e.g., (1b) 3. N is ppulatin size; f and F, functins f N. 4. Remarks. In Eq 1a, time is cntinuus. Slutins are curves. In Eq 1b, time is discrete, e.g., generatin number. Slutins are sequences f pints, e.g.,,,. 6

7 B. Clsed Ppulatins. 1. Ignre I and E. Then Cntinuus case. / (2a) Discrete case. (2b) N is ppulatin size (density); t, generatin number, etc. 1. Remarks. In the cntinuus case, the units f B and D are individuals (animals, plants, bacteria, etc.) per unit time. In the discrete case, the units f B and D are individuals. 7

8 C. Mdel Frmulatin. 1. Replace ppulatin parameters, B and D with crrespnding per capita quantities, 2. Fr the cntinuus case, / ; / (3a) 3. Fr the discrete case, it is cnvenient t define the per individual survival prbability, 1. Then 1 (3b) 4. Remarks. Units f r are (1/time), e.g., 1/s; R is dimensinless. 8

9 D. Slutins. 1. Cntinuus case: (4a) Ppulatin either grws (r > 0) r cllapses (r < 0). r = 0 is the watershed between grwth and cllapse. 2. Discrete case: (4b) Ppulatin either grws (R> 1) r cllapses (R < 1). Slutins t Eq. 3a fr r > 0. The figure suggests that is 0. Why is this impssible? What wuld the crrespnding slutins t Eq 3b lk like? R = 1 is the watershed between grwth and cllapse. 3. Ntice that 9

10 E. Dubling time (T 2 ) 1. Cntinuus case: Frm Eq (4a): Slve Yields 2 ln 2/.693/ (5a) 2. Discrete case: Frm Eq (4b): Slve 2 Take lgarithms f bth sides and slve. ln2/ln.693/ln (5b) 10

11 II. Equilibria and their Stability. A. Equilibria. Cnsider / (1a) (1b) 1. N* is an equilibrium f Eq 1a if 0 (6a) 2. N* is an equilibrium f Eq 1b if (6b) 3. In wrds: N* is an equilibrium f (1a) r (1b) if, absent external perturbatins, N des nt change. Fr Eqs 3a and 3b, the nly equilibrium is N = 0 prvided 0 ( 1). 4. If r = 0 r R=1, all values f N are equilibria. 11

12 5. Terminlgy: N is called a variable (ften called a state variable) r and R are called parameters. 12

13 B. Stability. 1. Refers t the system s respnse t a perturbatin. If fllwing a small perturbatin, the system returns t equilibrium, we say the equilibrium is stable. If fllwing a small perturbatin, the system mves further away frm equilibrium, the equilibrium is unstable. If the system neither returns t nr mre mves further away frm equilibrium, we say that the equilibrium is neutrally stable. 2. Freging cncept mre accurately referred t as lcal stability. Applies t the system s behavir near equilibrium. Fr expnential and gemetric grwth, there is nly ne equilibrium, 0. Mre generally there can be multiple equilibria. 3. A glbally stable system has the prperty that all initial states tend t a single equilibrium precludes existence f multiple equilibria. 13

14 Fr Eq 3a, N*=0, is unstable if r>0; stable if r<0. 14

15 C. Terminlgy. 1. A bifurcatin is a change in a system s qualitative behavir cnsequent t an arbitrarily small change in a parameter. 2. The parameter values at which bifurcatins ccur are called bifurcatin pints, bifurcatin values, etc. 3. In the case f expnential and gemetric grwth, the bifurcatin pints are 0 and 1. a. Fr 0 ( 1), an initial deviatin,, frm equilibrium shrinks. b. Fr 0 ( 1), increases. 15

16 D. Effect f Cnstant Remval (Emigratin, Harvesting)., 0 (7) 1. N* unstable, but shifted t the right. Ppulatin blws up if N 0 > N*. Otherwise, ppulatin cllapses. With emigratin r cnstant harvesting, the ppulatin expldes if N 0 >N* and cllapses if N 0 <N*. 16

17 E. Maintenance f a Cllapsing Ppulatin by Cnstant Immigratin., 0 (8) 1. N* stable, but shifted t the right. N(t) N*. Immigratin can maintain an therwise cllapsing ppulatin, In this case, N* is stable. 17

18 III. Randm Perturbatins. 1. Mdels we have been discussing are deterministic enter the same initial cnditins (N 0 in the cases cnsidered) and yu get the same answer every time. 2. But real wrld ppulatins (even thse in the lab) are subject t randm perturbatins. One can argue whether r nt Gd plays dice with the universe i.e., if anything is truly randm. But As a practical matter, it is ften apprpriate t imagine that lts f things are. 3. Tw basic surces f randmness. Intrinsic s-called demgraphic stchasticity. c. The average per-capita birth rate may be b, but nt all individuals prduce that many ffspring. d. Instead there is a prbability distributin functin (PDF). Extrinsic envirnmental perturbatins can be mdeled as fluctuating parameter values. 18

19 IV. Limits t Grwth. A. Expnential grwth can t g n frever. Eventually, smething becmes limiting. Limiting factrs include 1. Waste prduct / txin accumulatin. 2. Resurce depletin. 3. Increased predatin, parasitism and disease. 4. Increased aggressin. B. In the case f the human species, wrld ppulatin increasing at a less than expnential rate. 1. Hw much less unclear. 2. Depends n future human a. Reprductive behavir. b. Mrtality rates. Wrld ppulatin past and prjected. Three UN (2010) prjectins reflect differing fertility scenaris. Absent declining fertility, wrld ppulatin wuld rise t abut 26 billin by

20 C. Malthusian Trap. Human ppulatin grwth ver the past 12,000 years (left) has been predicted t presage wrld-wide catastrphe as the demand fr fd and ther resurces utstrips sciety s ability t prduce them. Equality f resurce prductin and cnsumptin is smetimes referred t as the Malthusian trap. 1. Thmas Malthus ( ) argued that ppulatin size increases expnenetinally, whereas fd prductin increases arithmetically. 2. Accrding t this, fd (and ther resurce) requirements will eventually exceed prductin thereby summning the hrsemen f the apcalypse and ppulatin cllapse. 20

21 D. Demgraphic Transitin. 1. Histrically, ecnmic develpment accmpanied by redutins in mrtality and fertility. Typically mrtality reductins lead. Ppulatin grwth first increases, then declines. 2. Immediate cnsequences: Shifts age structure t lder age classes. Increased wrker t dependent (mstly children) rati. Tp. Demgraphic transitin Bttm. Changing Japanese fertility. Increased per capita prductivity cntributed t the Asian miracle because mre f the capitas were wrking. 21

22 Changing ppulatin structure in Germany frm 1910 t Lng-term cnsequences: Reduced wrker t dependent (mstly ldsters) rati. Reduced prductivity 4. Demgraphic transitin a respnse t changing mres. Malthusian catastrphe thus far frestalled by technlgical advance. Meanwhile fertility has drpped belw replacement levels in many first wrld cuntries s-called Secnd Demgraphic Transitin. 22