jos e miguel ponciano: josemi@u .edu university of florida, … · 2016. 2. 15. · jos e miguel...

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Chapter 12: Population dynamics I: growth andregulation

José Miguel Ponciano: [email protected]

University of Florida, Biology Department

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What’s coming this week (and Tuesday of next week)

• Chapter 12, 13

• Some of the contents of chapter 11 and a bit of 10 will be mentioned in between thematerial for chapters 12 and 13.

• I will post review questions for the exam on Friday the 19th

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Life cycle graph: youngs and adults

n1σ1(1 − γ) n2 σ2σ1γ

φ

4


n1σ1(1 − γ) n2 σ2σ1γ

φ

n1(t + 1) = n1(t)

5


n1σ1(1 − γ) n2 σ2σ1γ

φ

n1(t + 1) = σ1︸︷︷︸prob. surv.

×n1(t)

6


n1σ1(1 − γ) n2 σ2σ1γ

φ

n1(t + 1) = σ1(1− γ)︸︷︷︸prob. surv. & not matur.

×n1(t)

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n1σ1(1 − γ) n2 σ2σ1γ

φ


×n1(t) +︸︷︷︸AND

8


n1σ1(1 − γ) n2 σ2σ1γ

φ


×n1(t) +︸︷︷︸AND

n2(t)

9


n1σ1(1 − γ) n2 σ2σ1γ

φ


×n1(t) +︸︷︷︸AND

φ︸︷︷︸# of offspring

×n2(t)

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n1σ1(1 − γ) n2 σ2σ1γ

φ


×n1(t) +︸︷︷︸AND

φ︸︷︷︸# of offspring

×n2(t)

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n1σ1(1 − γ) n2 σ2σ1γ

φ

n1(t + 1) = σ1(1− γ)n1(t) + φn2(t)

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n1σ1(1 − γ) n2 σ2σ1γ

φ

n1(t + 1) = σ1(1− γ)n1(t) + φn2(t)

n2(t + 1) = n1(t)

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n1σ1(1 − γ) n2 σ2σ1γ

φ

n1(t + 1) = σ1(1− γ)n1(t) + φn2(t)

n2(t + 1) = σ1γ × n1(t)︸︷︷︸# surv. and mat. juvs.

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n1σ1(1 − γ) n2 σ2σ1γ

φ

n1(t + 1) = σ1(1− γ)n1(t) + φn2(t)


+︸︷︷︸AND

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n1σ1(1 − γ) n2 σ2σ1γ

φ

n1(t + 1) = σ1(1− γ)n1(t) + φn2(t)


+︸︷︷︸AND

σ2 × n2(t)︸︷︷︸surv. adults

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n1σ1(1 − γ) n2 σ2σ1γ

φ

n1(t + 1) = σ1(1− γ)n1(t) + φn2(t)

n2(t + 1) = σ1γn1(t) + σ2n2(t)

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Model 1: Precocious and semelparous

n1σ1(1 − γ) n2 σ2σ1γ

φ

σ2 → 0, γ → 1

n1(t + 1) = σ1(1− γ)n1(t) + φn2(t)

n2(t + 1) = σ1γn1(t) + σ2n2(t)

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Precocious and Semelparous

Juveniles Adults Juv. Surv. Adult surv. Matur. Rate # offspring per adultYear n1 n2 sigma1 sigma 2 gamma phi

0 0 2 0.65 0.05 0.99 31 6 0.12 0.339 3.8663 11.6002035 0.41144654 1.30974082 7.485303285 22.4644231 1.217083386 3.7972689 14.51671057 43.5748136 3.169378068 9.79137047 28.19886159 84.6602284 7.7106899710 23.6823614 54.864391411 164.74711 17.982819112 55.0193136 106.91390613 321.099344 40.750623614 124.339017 208.66495915 626.80308 90.445405116 275.410435 407.87005217 1225.40032 197.62011818 600.825455 798.42611519 2399.18371 426.55248620 1295.25215 1565.2023421 4704.02616 911.75487722 2765.8408 3072.6285823 9235.86371 1933.4499924 5860.38307 6039.9507925 18157.9449 4073.1540526 12337.4888 11888.295227 35745.0794 8533.5887928 25833.1094 23428.63829 70453.8292 17795.0378 0

10000

20000

30000

40000

50000

60000

70000

80000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

n1

n2

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Model 2: precocious and iteroparous

n1σ1(1 − γ) n2 σ2σ1γ

φ

σ2 > 0, γ → 1

n1(t + 1) = σ1(1− γ)n1(t) + φn2(t)

n2(t + 1) = σ1γn1(t) + σ2n2(t)

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Precocious and iteroparous


0 0 2 0.65 0.85 0.99 31 6 1.72 5.139 5.3063 15.9514035 7.81704654 23.5548236 16.90921775 50.8807594 29.5303646 88.921817 57.84257817 174.105726 106.3873818 320.293829 202.4663089 609.480835 378.20544110 1138.57795 713.67554211 2148.42738 1339.2991212 4031.86214 2520.9172713 7588.95892 4737.2829714 14261.1771 8910.1855915 26823.2544 16750.725216 50426.5269 31498.880717 94824.4145 59223.518618 178286.915 111359.50219 335237.37 209383.20620 630328.66 393700.97221 1185200.05 740262.31922 2228490.76 1391899.2123 4190182.81 2617148.1324 7878680.57 4920958.5425 14814087.1 9252745.7126 27854528.7 17397698.927 52374151.1 32712433.328 98477731.8 61508334.529 185165109 115652505 0

20000000

40000000

60000000

80000000

100000000

120000000

140000000

160000000

180000000

200000000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

n1

n2

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Model 3: Delayed reproduction and semelparous

n1σ1(1 − γ) n2 σ2σ1γ

φ

σ2 → 0, γ < 1

n1(t + 1) = σ1(1− γ)n1(t) + φn2(t)

n2(t + 1) = σ1γn1(t) + σ2n2(t)

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Delayed reprod and semelparous


0 0 2 0.65 0.05 0.12 31 6 0.12 3.732 0.4733 3.553704 0.3147464 2.97695669 0.292926215 2.58159786 0.246848936 2.21722077 0.213707087 1.90937152 0.183628578 1.64304623 0.158112419 1.41415967 0.1360632310 1.21708901 0.1171076211 1.04749776 0.1007883212 0.90153369 0.0867442413 0.77590999 0.0746568414 0.66779104 0.0642538215 0.57473794 0.0553003916 0.49465128 0.0475945817 0.42572427 0.0409625318 0.36640187 0.0352546219 0.31534572 0.0303420820 0.27140398 0.0261140721 0.23358529 0.0224752122 0.20103643 0.0193434123 0.17302308 0.0166480124 0.14891324 0.014328225 0.12816297 0.0123316426 0.11030415 0.0106132927 0.09493385 0.0091343928 0.08170533 0.0078615629 0.07032013 0.00676609 0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

n1

n2

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Model 4: Delayed reproduction and iteroparous

n1σ1(1 − γ) n2 σ2σ1γ

φ

σ2 > 0, γ < 1

n1(t + 1) = σ1(1− γ)n1(t) + φn2(t)

n2(t + 1) = σ1γn1(t) + σ2n2(t)

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Delayed reprod and iteroparous


0 0 2 0.65 0.85 0.05 31 6 1.72 8.805 1.643 10.3570875 1.68016254 11.435989 1.764743475 12.3559536 1.871701596 13.2449061 1.992514857 14.1562741 2.124097078 15.1137905 2.265561429 16.1294499 2.4169253910 17.2107115 2.578593711 18.3633954 2.7511527712 19.592855 2.9352902113 20.9044586 3.1317644614 22.3037966 3.341394715 23.7967785 3.5650588816 25.3896874 3.8036953517 27.089218 4.0583058918 28.9025098 4.3299595919 30.8371786 4.6197972220 32.9013494 4.9290359421 35.1036911 5.258974422 37.4534525 5.610998223 39.9605015 5.9865856824 42.6353667 6.3873141225 45.4892813 6.8148664226 48.5342305 7.271038127 51.7830016 7.7577448828 55.2492381 8.277030729 58.9474967 8.83107633 0

10

20

30

40

50

60

70

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

n1

n2

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Model 5: A bit of realism: adult survival dependent ontemperature

n1σ1(1 − γ) n2 σ2σ1γ

φ

x(t) = temperature at time t

σ2becomesσ2(x(t)), γ < 1

n1(t + 1) = σ1(1− γ)n1(t) + φn2(t)

n2(t + 1) = σ1γn1(t) + σ2(x(t))n2(t)

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Narrow range of temperature survival

Temp. (x) Adult surv (sigma2) Select. Strength optimum temp31 3.78028E-‐27 0.08 7032 8.22528E-‐2634 3.06319E-‐2336 8.28368E-‐2138 1.62666E-‐1840 2.31952E-‐1642 2.40173E-‐1444 1.80583E-‐1246 9.85951E-‐1148 3.90894E-‐0950 1.12535E-‐0752 2.35258E-‐0654 3.57128E-‐0556 0.00039366958 0.00315111260 0.01831563962 0.0773047464 0.23692775966 0.52729242468 0.85214378970 172 0.85214378974 0.52729242476 0.23692775978 0.0773047480 0.01831563982 0.00315111284 0.00039366986 3.57128E-‐0588 2.35258E-‐0690 1.12535E-‐0792 3.90894E-‐0994 9.85951E-‐1196 1.80583E-‐1298 2.40173E-‐14

100 2.31952E-‐16102 1.62666E-‐18104 8.28368E-‐21106 3.06319E-‐23108 8.22528E-‐26110 1.60381E-‐28

-‐0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120

Adult surv (sigma2)

Adult surv (sigma2)

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Population trend with narrow range of temperaturesurvival

Temperature-‐dependent survival of adults

Juveniles Adults Temp Adult surv. Juv. Surv. Matur. Rate # offspring per adultYear n1 n2 x(t) sigma2(x(t)) sigma1 gamma phi

0 0 2 99 2.4566E-‐15 0.65 0.12 31 6 4.9132E-‐15 63 0.1408584212 3.432 0.468 86 3.57128E-‐053 3.367104 0.26771271 108 8.22528E-‐264 2.72912163 0.26263411 38 1.62666E-‐185 2.34895991 0.21287149 86 3.57128E-‐056 1.98221953 0.18322648 75 0.3678794417 1.683509 0.22201838 60 0.0183156398 1.62902227 0.13538011 51 5.35535E-‐079 1.33794107 0.12706381 64 0.23692775910 1.14649372 0.13446435 50 1.12535E-‐0711 1.05918745 0.08942653 98 2.40173E-‐1412 0.8741348 0.08261662 42 2.40173E-‐1413 0.74785497 0.06818251 58 0.00315111214 0.63232058 0.05854754 59 0.00790705415 0.53732999 0.04978394 44 1.80583E-‐1216 0.45670459 0.04191174 58 0.00315111217 0.38697024 0.03575503 68 0.85214378918 0.32861206 0.0606521 58 0.00315111219 0.3699224 0.02582286 32 8.22528E-‐2620 0.2890642 0.02885395 31 3.78028E-‐2721 0.25190657 0.02254701 34 3.06319E-‐2322 0.21173158 0.01964871 84 0.00039366923 0.1800566 0.0165228 98 2.40173E-‐1424 0.15256077 0.01404441 51 5.35535E-‐0725 0.129398 0.01189975 88 2.35258E-‐0626 0.1097149 0.01009307 42 2.40173E-‐1427 0.09303614 0.00855776 104 8.28368E-‐2128 0.07888996 0.00725682 51 5.35535E-‐0729 0.06689551 0.00615342 110 1.60381E-‐28

0

1

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3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

n1

n2

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Broad range of temperature survival

Temp. (x) Adult surv (sigma2) Select. Strength optimum temp31 0.022314915 0.005 7032 0.02705184734 0.03916389536 0.05557621338 0.0773047440 0.10539922542 0.14085842144 0.18451952446 0.23692775948 0.29819727950 0.36787944152 0.44485806654 0.52729242456 0.61262639458 0.69767632660 0.77880078362 0.85214378964 0.91393118566 0.96078943968 0.99004983470 172 0.99004983474 0.96078943976 0.91393118578 0.85214378980 0.77880078382 0.69767632684 0.61262639486 0.52729242488 0.44485806690 0.36787944192 0.29819727994 0.23692775996 0.18451952498 0.140858421

100 0.105399225102 0.07730474104 0.055576213106 0.039163895108 0.027051847110 0.018315639

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120

Adult surv (sigma2)

Adult surv (sigma2)

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Population trend with broad range of temperaturesurvival

Temperature-‐dependent survival of adults

Juveniles Adults Temp Adult surv. Juv. Surv. Matur. Rate # offspring per adultYear n1 n2 x(t) sigma2(x(t)) sigma1 gamma phi

0 0 2 53 0.485536895 0.65 0.12 31 6 0.97107379 62 0.8521437892 6.34522137 1.2954945 95 0.2096113873 7.51595012 0.76647767 60 0.7788007834 6.59855647 1.18317752 69 0.9975031225 7.32390685 1.69491067 57 0.6554062546 9.27400673 1.68211979 69 0.9975031227 10.3510912 2.40129227 60 0.7788007838 13.124701 2.67751341 36 0.0555762139 15.5398692 1.17253273 51 0.40555450510 12.4064034 1.68763573 46 0.23692775911 12.1593699 1.36754721 36 0.05557621312 11.0578012 1.02443395 102 0.0773047413 9.39836415 0.9417021 39 0.09049144214 8.20097058 0.81828838 64 0.91393118515 7.14582032 1.38753498 48 0.29819727916 8.25001416 0.97113314 42 0.14085842117 7.63240752 0.78029339 75 0.93941306318 6.70661726 1.32834559 53 0.48553689519 7.82122183 1.16807694 69 0.99750312220 7.9779697 1.77521569 98 0.14085842121 9.88904575 0.87233572 98 0.14085842122 8.27354132 0.8942214 81 0.73896848823 7.41512983 1.30613766 97 0.16162119224 8.15986724 0.78947965 90 0.36787944125 7.03588302 0.92690298 39 0.09049144226 6.80523402 0.63267566 65 0.93941306327 5.79062085 1.12515204 95 0.20961138728 6.68769123 0.68751311 77 0.88470590529 5.8878987 1.12988682 82 0.697676326

0

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

n1

n2

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What is ‘density-dependence’?

• Book def: “Density dependent” Factors that affect population size in relation to thepopulation’s density

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• Embedded in this definition is the idea that somehow, the population density (or size, ingeneral) affects changes in population size.

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• The idea of density dependence is very general and is not associated with a singlemechanism by which the future growth of a population is affected via the current

population size (or past population sizes).

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• Because of its generality, the concept has been very controversial among ecologists, who, asbiologists, are eager to clarify the specific mechanism behind the connection between

current population sizes and the growth rate of a particular population

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• Because of its generality, the concept has been very controversial among ecologists, who, asbiologists, are eager to clarify the specific mechanism behind the connection between

current population sizes and the growth rate of a particular population

• The problem is that density-dependence is usually advanced as a general description of theexpected pattern of change in births and deaths when these are formulated as a function of

population density (size). Often, little attention is put, however, to the mechanism

resulting in, say, increased deaths at higher population sizes. See discussion about

density-dependence in B. Dennis and M.L. Taper 1994, Ecological monographs

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A notable exception: Ricker 1954

Haccou et al 2005. Branching Processes. Mortality in non-overlapping generations of fish. zn+1 = znme−bzn = zne

a−bzn , where m = ea.

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The Ricker model of density-dependent populationgrowth

Let Nt be population size at time t:

Nt+1 = Ntm = Nt exp [a], wherem = exp [a].

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

250

300

350

400

450

Time, in Days

Para

mecia

density(ind./.5

cc)

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Nt+1 = Nt exp [a + bNt]

0 5 10 15 200

50

100

150

200

250

300

350

400

450

Time, in Days

Pa

ram

ecia

de

nsity (

ind

./.5

cc)

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0 5 10 15 200

50

100

150

200

250

300

350

400

450

Time, in Days

Para

mecia

density (

ind./.5

cc)

!!"#

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Why do we see these departures of real data fromdeterministic predictions?



0 5 10 15 200

50

100

150

200

250

300

350

400

450

Time, in Days

Pa

ram

ecia

de

nsity (

ind

./.5

cc)

G.F. Gause, 1934. “The struggle for existence”.

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Other examples

Chaos in Ecology, 2002 (Cushing et al). Flour beetles, a common pest of various important

crops, cannibalize their young. Larvae eat eggs and adults eat eggs and pupae. The greater

the densities of adults, the greater the death rate of pupae and eggs by adults. Again, the

natural process that ends up in deaths that depend on densities, are a function of an

exponential term of the form

e−constant×number or density of adults.

Webpage: http://caldera.calstatela.edu/nonlin/index.html.

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Other examples








Another relevant mechanism of death that becomes more prevalent as population size (or

density) increase is parasitism. Nicholson (1933) and Nicholson and Bailey (1935), derived

a model where as the density of the parasitoid increased, the probability that a host would

escape parasitism decreased.

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Other examples








Another relevant mechanism of death that becomes more prevalent as population size (or

density) increase is parasitism. Nicholson (1933) and Nicholson and Bailey (1935), derived

a model where as the density of the parasitoid increased, the probability that a host would

escape parasitism decreased.

There is literally a multitude of mechanisms that would produce either an increase in deaths as

population size grows or a decrease in births.

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Other examples

Nowadays, the exponential form shown above is typically used to indicate density-dependence,

but it is definitely NOT the only way of creating ‘density-dependent’ models. I do prefer

models where the mechanism is explicit.

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Model 6: Density-dependent reproduction

n1σ1(1 − γ) n2 σ2σ1γ

φ

Let N(t) = n1(t) + n2(t). Then φ(N(t)) = φe−bN(t)

n1(t + 1) = σ1(1− γ)n1(t) + φe−bN(t)n2(t)

n2(t + 1) = σ1γn1(t) + σ2n2(t)

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Model 7: Density-dependent maturation

n1σ1(1 − γ) n2 σ2σ1γ

φ

Let N(t) = n1(t) + n2(t). Then γ(N(t)) = γe−bN(t)

n1(t + 1) = σ1(1− γe−bN(t))n1(t) + φn2(t)

n2(t + 1) = σ1γn1(t) + σ2n2(t)

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Model 8: Density-dependent juvenile survival

n1σ1(1 − γ) n2 σ2σ1γ

φ

Let N(t) = n1(t) + n2(t). Then σ1(N(t)) = σ1e−bN(t)

n1(t + 1) = σ1e−bN(t)(1− γ)n1(t) + φn2(t)

n2(t + 1) = σ1e−bN(t)γn1(t) + σ2n2(t)

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Model 9: Density-dependent adult survival

n1σ1(1 − γ) n2 σ2σ1γ

φ

Let N(t) = n1(t) + n2(t). Then σ1(N(t)) = σ1e−bN(t)

n1(t + 1) = σ1(1− γ)n1(t) + φn2(t)

n2(t + 1) = σ1γn1(t) + σ2e−bN(t)n2(t)

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Density dependent models in Excel spreadsheet

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Ecology as a general science to understand naturalhistories

Instead of learning about a particular natural system (either fish, reptiles, amphibians,

mammals, plants, invertebrates, fungi, etc.), ecological thinking is meant to bring

understanding through generality.

35





Patterns in nature are understood through hypotheses that get at the processes generating

these patterns

36






these patterns

Ecological models can explain observed patterns of population trends by phrasing hypotheses in

simple mathematical terms

37






these patterns

Ecological models can explain observed patterns of population trends by phrasing hypotheses in

simple mathematical terms

Mathematical models are amenable to testing, through data in a way very similar to the way

you test whether the slope in a simple linear regression is significant or not

38

Next time:

• Exponential growth and the logistic growth equation

• The Allee effect

• Fitting and comparing different models with real data: a very simple example

jos e miguel ponciano: josemi@u .edu university of florida, … · 2016. 2. 15. · jos e miguel...

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