-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathmidterm1.txt
1757 lines (933 loc) · 41.1 KB
/
midterm1.txt
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
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
UNIT Midterm 1 extra notes
----------------------------------------------------------------------
TSEC Introduction
In \textbf{linear} population models, per capita rates are independent of
population size
In \textbf{non-linear} models, not so much
Why might per capita birth and death rates change with population size?
What does this imply about population \textbf{dynamics}
----------------------------------------------------------------------
The first law of population dynamics
If individuals are behaving independently:
the population-level rate of growth (or decline) is proportional
to the population size
the population grows (or declines) exponentially
----------------------------------------------------------------------
The second law of population dynamics
Exponential growth (or decline) cannot continue forever
Something is changing the average rate at which populations we
observe grow
----------------------------------------------------------------------
PICSLIDE
FIG webpix/ecoli.jpg
----------------------------------------------------------------------
The third law of population dynamics
Exponential growth (or decline) cannot continue forever --
\emph{even on average}
Environmental variation cannot be the only thing that changes growth
rates
Populations are, directly or indirectly, limiting their own growth
rates
This is called \textbf{density dependence}
----------------------------------------------------------------------
Long-term growth rates
Populations maintain long-term growth rates very close to $r=0$
This is almost certainly because factors affecting their growth rate
change with the size of population.
----------------------------------------------------------------------
Changing growth rates
POLL What is an example of a density-dependent mechanism that
affects growth rate?
ANS Predators and diseases
ANS As populations go up, pressure from natural enemies could
go up \emph{even faster}
ANS If pressure increases at the same rate, per capita effects could
stay the same
ANS Insufficient resources
ANS Limitation: e.g., oak trees use all the available light
ANS Destruction: gypsy moths kill all the oak trees
----------------------------------------------------------------------
Population regulation
All the populations we see are \emph{regulated}
On average, population growth is higher when the population is
lower
Maybe with a time delay
Why is this interesting?
Lots of populations don't \emph{look like} they are regulated
----------------------------------------------------------------------
Sometimes regulation is apparent
BC
Some species seem to fill a niche (mangroves)
or deplete their own food resources (gypsy moths)
NC
SIDEFIG webpix/mangroves.jpg
SIDEFIG webpix/gm_defoliation.jpg
EC
----------------------------------------------------------------------
Sometimes regulation is not apparent
BC
Other species seem like they could easily be more common (pine trees)
May be controlled by cryptic (hard to see) natural enemies (like
disease or parasites)
May be controlled by limitations that occur only at certain times
(e.g., during regular droughts)
NC
SIDEFIG webpix/forest.jpg
EC
----------------------------------------------------------------------
Regulation works over the long term
Not every species is experiencing population regulation at every
time
A species that we see now may be expanding into a niche (e.g.,
because of climate change)
Some species are controlled by big outbreaks of disease
Some species have big outbreaks into marginal habitat, and spend
most of their time contracting back to their ``core" habitat
----------------------------------------------------------------------
How do we know it's regulation?
POLL Why don't we believe that population growth is controlled by factors that don't depend on the population itself?
ANS Because the long-term average value of $r$ has to be
\emph{very} close to 0
ANS This is true for \emph{every} population
ANS This is unlikely to occur by chance
ANS Thus, it must be through direct or indirect responses to the
population size
----------------------------------------------------------------------
SS Population Examples
----------------------------------------------------------------------
PRESLIDE
Elk
DOUBLEPDF ts/elk.elkplot.Rout
----------------------------------------------------------------------
CSLIDE Ebola
DOUBLEFIG WA_Ebola_Outbreak/liberia.npc.tsplot.Rout-0.pdf WA_Ebola_Outbreak/liberia.npc.tsplot.Rout-1.pdf
----------------------------------------------------------------------
PRESLIDE Paramecia
DOUBLEPDF ts/paramecia.plot.Rout
----------------------------------------------------------------------
PRESLIDE Gypsy moths
DOUBLEPDF ts/gm10165.plot.Rout
----------------------------------------------------------------------
Gypsy moths
POLL What are some factors that limit gypsy-moth populations?
Which are likely to be affected by the moths?
Directly or indirectly, in the short or long term?
----------------------------------------------------------------------
SEC Continuous-time regulation
----------------------------------------------------------------------
Build on the linear model
Our linear population model is:
$\bigeq \frac{dN}{dt} = (b-d)N $
Per-capita rates are constant
\textbf{Population-level rates are linear}
Behaviour is exponential
----------------------------------------------------------------------
Individual perspective
DOUBLEFIG bd_models/constant.bd.Rout-0.pdf bd_models/constant.bd.Rout-2.pdf
----------------------------------------------------------------------
CONT Individual perspective
BC
Per capita rate shows birth and death per individual
Corresponds to the time plot showing growth on a log scale
On the log scale we see \emph{multiplicative} or \emph{proportional} change
NC
SIDEFIG bd_models/constant.bd.Rout-0.pdf
SIDEFIG bd_models/constant.bd.Rout-2.pdf
EC
----------------------------------------------------------------------
Population perspective
DOUBLEFIG bd_models/constant.bd.Rout-1.pdf bd_models/constant.bd.Rout-3.pdf
----------------------------------------------------------------------
CONT Population perspective
BC
Total rate shows birth and death for the whole population
Corresponds to the time plot showing growth on a linear scale
On the linear scale we see \emph{additive} or \emph{absolute}
change
NC
SIDEFIG bd_models/constant.bd.Rout-1.pdf
SIDEFIG bd_models/constant.bd.Rout-3.pdf
EC
----------------------------------------------------------------------
Non-linear model
Population has \emph{per capita} birth rate $b(N)$ and death
rate $d(N)$
Per-capita rates change with the population size
Our non-linear model is: $\bigeq \frac{dN}{dt} = (b(N)-d(N))N$
$\equiv r(N) N$
Defines how fast the population is changing at any instant
----------------------------------------------------------------------
Recruitment
{\bf Recruitment} is when an organism moves from one life stage to
another:
Seed \rec seedling \rec sapling \rec tree
Egg \rec larva \rec pupa \rec moth
In simple continuous-time population models, recruitment is included
in birth:
$b$ is the rate at which adults produce new adults; or seeds
produce new seeds -- we have to ``close the loop"
----------------------------------------------------------------------
Birth rates
BC
When a population is crowded, the birth rate will usually go down
Resources are limited: space, food, light
But it may stay the same
Or even go up
If individuals shift their resources to reproduction instead of
survival
NC
SIDEFIG webpix/mussels.jpg
SIDEFIG webpix/fir_wave.jpg
EC
----------------------------------------------------------------------
Death rates
BC
When a population is crowded, the death rate will often go up
Individuals are starving, or conflict increases
But it may stay the same
if reproduction is limited by competition for breeding sites,
or by recruitment of juveniles
Or even go down
if organisms go into some sort of ``resting mode"
NC
SIDEFIG webpix/seagulls.jpg
SIDEFIG webpix/spores.png
EC
----------------------------------------------------------------------
Reproductive numbers
Our model is: $\bigeq \frac{dN}{dt} = (b(N)-d(N))N$
$\equiv r(N) N$
Reproductive number now also changes with $N$:
ANS $\R(N) = b(N)/d(N)$
When the population is crowded, individuals are stressed and
the reproductive number will typically go down.
----------------------------------------------------------------------
Carrying capacity
If a population has $\R(N) > 1$ when it's not crowded
The population should increase
Eventually, \R\ will decrease, and eventually cross $\R=1$
We call the special value of $N$ where $\R(N) = 1$, the
\textbf{carrying capacity}, $K$
$\R(K) \equiv 1$
$b(K) \equiv d(K)$
When $N=K$:
ANS Population stays the same, on average
----------------------------------------------------------------------
Logistic model
A popular model of density-dependent growth is the logistic model
Per capita instantaneous growth rate $r$ is a function of $N$
$r(N) = \rmax(1-N/K)$
Consistent with various assumptions about $b(N)$ and $d(N)$
Population increases to $K$ and remans there
Units of $N$ must match units of $K$
Not a linear model, because \emph{population-level} rates are not
linear
----------------------------------------------------------------------
Exponential-rates model
In this course, we'll mostly use another simple model:
$b(N) = b_0 \exp(-N/N_b)$
$d(N) = d_0 \exp(N/N_d)$
This is the simplest model that is smooth and keeps track
of birth and death rates separately
Birth rate goes down with characteristic scale $N_b$
Death rate goes up with characteristic scale $N_d$
----------------------------------------------------------------------
Exponential-rates vs.\ logistic
The exponential-rates model is conceptually clearer
Birth and death rates are clearly defined
Mathematically nicer
Rates always stay positive
The logistic \emph{looks} less scary
----------------------------------------------------------------------
SS A simple, continuous-time model
----------------------------------------------------------------------
Assumptions
We model individual-level rates, but individuals are \emph{not}
independent: my rates depend on the number (or density) of
individuals in the population
The population can be censused at any time
Population size changes continuously
All individuals are the same all the time
Population changes deterministically
----------------------------------------------------------------------
Interpretation
If we have $N$ individuals at time $t$, how does the population
change?
Individuals are giving birth at per-capita rate $b(N)$
Individuals are dying at per-capita rate $d(N)$
Population dynamics follow:
$\bigeq \frac{dN}{dt} = (b(N)-d(N))N \equiv r(N) N$
----------------------------------------------------------------------
States and state variables
What variable or variables describe the state of this system?
ANS The same as before: population size (or density)
ANS We are still assuming that's all we need to know
ANS In other words, that all individuals are the same.
----------------------------------------------------------------------
Parameters
POLL What quantities describe the rules for this system?
ANS $b_0$ [1/time]
ANS $d_0$ [1/time]
ANS $N_b$ [indiv] (or [indiv/area])
ANS $N_d$ [indiv] (or [indiv/area])
----------------------------------------------------------------------
Characteristic \emph{scale}
A characteristic scale for density dependence is analogous to a
characteristic time
For example: $b(N) = b_0 \exp(-N/N_b)$
$N_b$ is the characteristic scale of density-dependence in birth
rate
If $N \ll N_b$, density dependence is linear (and relatively
small)
If $N \gg N_b$, density dependence is exponential, and very
large (virtually no births)
----------------------------------------------------------------------
Model
Dynamics:
$\bigeq \frac{dN}{dt} = (b_0 \exp(-N/N_b) - d_0 \exp(N/N_d))N$
Exact solution:
Insanely complicated
Behaviour of the solution:
Pretty easy!
----------------------------------------------------------------------
Dynamics
What sort of \textbf{dynamics} do we expect from our conceptual
model?
I.e., how will it change through time?
What will the population do if it starts
near zero?
near the equilibrium?
at a high value?
----------------------------------------------------------------------
PSLIDE What will this model do?
FIG bd_models/birth.bd.Rout-0.pdf
----------------------------------------------------------------------
PSLIDE What will this model do?
FIG bd_models/birth.bd.Rout-6.pdf
----------------------------------------------------------------------
PSLIDE What will this model do?
FIG bd_models/birth.bd.Rout-8.pdf
----------------------------------------------------------------------
What will this model do?
BCC
WIDEFIG bd_models/birth.bd.Rout-8.pdf
NCC
Increase when population is below equilibrium
Decrease when population is above equilibrium
Converge
EC
----------------------------------------------------------------------
PSLIDE Low starting population example
DOUBLEFIG bd_models/birth.bd.Rout-8.pdf bd_models/birth.bd.Rout-2.pdf
----------------------------------------------------------------------
PSLIDE High starting population example
DOUBLEFIG bd_models/birth.bd.Rout-8.pdf bd_models/birth.bd.Rout-4.pdf
----------------------------------------------------------------------
Examples
DOUBLEFIG bd_models/birth.bd.Rout-2.pdf bd_models/birth.bd.Rout-4.pdf
----------------------------------------------------------------------
SS Simulating model behaviour
----------------------------------------------------------------------
Simulations
We will simulate the behaviour of populations in continuous time
using the program R
This program generates the pictures in this section by implementing
our model of how the population changes instantaneously
----------------------------------------------------------------------
Individual-scale pictures
We can view graphs of our population assumptions on the
individual scale
per-capita birth and death rates
units [1/time]
what is each individual doing (on average)?
corresponds to the dynamics we visualize on a log-scale graph of
the population
NOTES See above
----------------------------------------------------------------------
REPSLIDE What will this model do?
DOUBLEFIG bd_models/birth.bd.Rout-8.pdf bd_models/birth.bd.Rout-2.pdf
----------------------------------------------------------------------
Population-scale pictures
We can view graphs of our population assumptions on the
population scale
total birth and death rates
units [indiv/time]
or [density/time] = [(indiv/area)/time]
what is changing at the population level?
corresponds to the dynamics we visualize on a linear-scale graph
of the population
----------------------------------------------------------------------
Population perspective picture
DOUBLEFIG bd_models/birth.bd.Rout-9.pdf bd_models/birth.bd.Rout-3.pdf
----------------------------------------------------------------------
Decreasing birth rate
BC
Decreasing birth rate (above) might be a good model for organisms
that experience density dependence primarily in the recruitment stage
For example, we might count adult trees, and these might not die
more at high density -- just fail to recruit younger ones
NC
SIDEFIG webpix/tongass.jpg
EC
----------------------------------------------------------------------
REPSLIDE Decreasing birth rates
FIG bd_models/birth.bd.Rout-0.pdf
----------------------------------------------------------------------
PSLIDE Increasing death rates
FIG bd_models/death.bd.Rout-0.pdf
----------------------------------------------------------------------
Increasing death rate
BC
Increasing death rate might be a good model for organisms
that experience density dependence primarily as adults
For example, in some environments, mussel density might be limited
by adult crowding. Although juvenile mussels tend to have a hard
time, this might not be density dependent
NC
SIDEFIG webpix/mussels.jpg
EC
----------------------------------------------------------------------
Individual perspective
DOUBLEFIG bd_models/death.bd.Rout-0.pdf bd_models/death.bd.Rout-2.pdf
----------------------------------------------------------------------
Population perspective
DOUBLEFIG bd_models/death.bd.Rout-1.pdf bd_models/death.bd.Rout-3.pdf
----------------------------------------------------------------------
Decreasing death rate
BC
Some organisms (such as many types of bacteria) slow down their
metabolisms under density dependence, so that death rate
\emph{decreases}
How is this consistent with density dependence?
ANS Birth rate must decrease even faster
NC
SIDEFIG webpix/spores.png
EC
----------------------------------------------------------------------
Individual perspective
DOUBLEFIG bd_models/slow.bd.Rout-0.pdf bd_models/slow.bd.Rout-2.pdf
----------------------------------------------------------------------
Population perspective
DOUBLEFIG bd_models/slow.bd.Rout-1.pdf bd_models/slow.bd.Rout-3.pdf
----------------------------------------------------------------------
Other examples
There are two other possible scenarios for density dependence
For fun, you can try to think of what they are
But all of these examples have similar behaviour
Increase from low density
Decrease from high density
Approach carrying capacity
----------------------------------------------------------------------
PRESLIDE Maximum growth rates
POLL When does a population in this model have the fastest
\emph{per-capita} growth rate? | When is per capita growth fastest?
NOANS When density is low.
NOANS This is an assumption.
POLL When does a population in this model have the fastest \emph{total}
growth rate? | When is total growth fastest?
NOANS Intermediate between low density and the carrying capacity.
NOANS This is something we learn from the model!
----------------------------------------------------------------------
REPSLIDE Individual perspective
DOUBLEFIG bd_models/birth.bd.Rout-0.pdf bd_models/birth.bd.Rout-2.pdf
----------------------------------------------------------------------
REPSLIDE Population perspective
DOUBLEFIG bd_models/birth.bd.Rout-1.pdf bd_models/birth.bd.Rout-3.pdf
----------------------------------------------------------------------
Maximum growth rates
When does a population in this model have the fastest
\emph{per-capita} growth rate?
ANS When density is low.
ANS This is an assumption.
When does a population in this model have the fastest \emph{total}
growth rate?
ANS Intermediate between low density and the carrying capacity.
ANS This is a something we learn from the model
----------------------------------------------------------------------
REPSLIDE Individual perspective
DOUBLEFIG bd_models/birth.bd.Rout-0.pdf bd_models/birth.bd.Rout-2.pdf
----------------------------------------------------------------------
REPSLIDE Population perspective
DOUBLEFIG bd_models/birth.bd.Rout-1.pdf bd_models/birth.bd.Rout-3.pdf
----------------------------------------------------------------------
TSS Equilibria and time scales
We define \textbf{equilibrium} as when the population is not changing
Our simple model is $\bigeq \frac{dN}{dt} = (b(N)-d(N))N$
In this simple model, when does equilibrium occur?
ANS $b(N) = d(N)$ (the carrying capacity)
ANS $N=0$ (the population is absent)
----------------------------------------------------------------------
Stable and unstable equilibria
Aren't equilibria always stable?
If we are at an equilibrium we expect to stay there
(in our simplified model, at least)
An equilibrium is defined as stable if we expect to approach the
equilibrium \emph{when we are near it.}
An equilibrium is defined as unstable if we expect to move away from
the equilibrium \emph{when we are near it.}
----------------------------------------------------------------------
REPSLIDE Population perspective
DOUBLEFIG bd_models/birth.bd.Rout-1.pdf bd_models/birth.bd.Rout-3.pdf
----------------------------------------------------------------------
What kind of equilibrium?
How can we tell an equilibrium is stable?
If population is just below the equilibrium:
ANS It should increase ($b>d$)
If population is just above the equilibrium:
ANS It should decrease ($d>b$)
----------------------------------------------------------------------
Basic reproductive number
The reproductive number of a population not affected by crowding is
called the \textbf{basic reproductive number}
Written \Ro\ or \Rmax.
In this model, when $\Ro<1$ the population:
ANS Always decreases
When $\Ro>1$ the population:
ANS Increases when it is small
ANS Eventually $\R$ will decrease
----------------------------------------------------------------------
CONT
POLL What is \Ro\ in our current model? | What is R0?
ANS $\Ro=b(0)/d(0)$
ANS $\Ro$, $b(0)$, and $d(0)$ are limits
ANS Nothing actually grows or dies when $N=0$
ADD A leading statement for the second question: how does that make sense?
----------------------------------------------------------------------
Invasion
We say a species can ``invade" a system if its rate of change is
positive when the population is small.
In other words, population can invade if the extinction equilibrium
is not stable
In this conceptual model, this is the same as saying $b(0) > d(0)$
Which is the same as saying $\Ro>1$
----------------------------------------------------------------------
Invasion examples
POLL What are some examples of biological invasions?
NOANS
NOANS
NOANS
NOANS
----------------------------------------------------------------------
Different behaviours
When $\Ro>1$, the population invades
Zero equilibrium is unstable, carrying capacity equilibrium is
stable
When $\Ro<1$, the population fails to invade
Zero equilibrium is stable, carrying capacity equilibrium does
not exist
----------------------------------------------------------------------
Individual perspective
DOUBLEFIG bd_models/fail.bd.Rout-0.pdf bd_models/fail.bd.Rout-2.pdf
When $\Ro<1$ population always decreases
----------------------------------------------------------------------
Population perspective
DOUBLEFIG bd_models/fail.bd.Rout-1.pdf bd_models/fail.bd.Rout-3.pdf
When $\Ro<1$ population always decreases
----------------------------------------------------------------------
REPSLIDE Individual perspective
DOUBLEFIG bd_models/birth.bd.Rout-0.pdf bd_models/birth.bd.Rout-2.pdf
When $\Ro>1$ population increases when it is small
----------------------------------------------------------------------
REPSLIDE Population perspective
DOUBLEFIG bd_models/birth.bd.Rout-1.pdf bd_models/birth.bd.Rout-3.pdf
When $\Ro>1$ population increases when it is small