946:
606:
650:
372:
941:{\displaystyle {\begin{aligned}{\begin{pmatrix}N_{t+l_{1}}\\N_{t+l_{2}}\\N_{t+l_{3}}\end{pmatrix}}&={\begin{pmatrix}F_{1}&F_{2}&F_{3}\\S_{1}&0&0\\0&S_{2}&0\end{pmatrix}}{\begin{pmatrix}N_{t_{1}}\\N_{t_{2}}\\N_{t_{3}}\end{pmatrix}}\end{aligned}}.}
615:
for this species. Each row in the first and third matrices corresponds to animals within a given age range (0–1 years, 1–2 years and 2–3 years). In a Leslie matrix the top row of the middle matrix consists of age-specific fertilities:
601:{\displaystyle {\begin{aligned}{\begin{pmatrix}N_{t+l_{i}}\\N_{t+l_{a}}\end{pmatrix}}&={\begin{pmatrix}S_{i}R_{i}&S_{a}R_{i}\\S_{i}&S_{a}\end{pmatrix}}{\begin{pmatrix}N_{t_{i}}\\N_{t_{a}}\end{pmatrix}}\end{aligned}}.}
321:
183:
Although BIDE models are conceptually simple, reliable estimates of the 5 variables contained therein (N, B, D, I and E) are often difficult to obtain. Usually a researcher attempts to estimate current abundance,
207:
For added simplicity it may help to think of time t as the end of the breeding season in year t and to imagine that one is studying a species that has only one discrete breeding season per year.
655:
377:
35:
of wildlife or human populations. Matrix algebra, in turn, is simply a form of algebraic shorthand for summarizing a larger number of often repetitious and tedious algebraic computations.
116:
962:
can be constants or they can be functions of environment, such as habitat or population size. Randomness can also be incorporated into the environmental component.
983:
Caswell, H. 2001. Matrix population models: Construction, analysis and interpretation, 2nd
Edition. Sinauer Associates, Sunderland, Massachusetts.
216:
994:
951:
These models can give rise to interesting cyclical or seemingly chaotic patterns in abundance over time when fertility rates are high.
988:
971:
1009:
1014:
192:
technique. Estimates of B might be obtained via a ratio of immatures to adults soon after the breeding season, R
48:
611:
Suppose that you are studying a species with a maximum lifespan of 4 years. The following is an age-based
640:
in the matrix above. Since this species does not live to be 4 years old the matrix does not contain an S
32:
197:
189:
28:
984:
20:
204:. Often, immigration and emigration are ignored because they are so difficult to estimate.
196:. Number of deaths can be obtained by estimating annual survival probability, usually via
362:= ratio of surviving young females at the end of the breeding season per breeding female
24:
1003:
612:
201:
180:
This equation is called a BIDE model (Birth, Immigration, Death, Emigration model).
39:
316:{\displaystyle N_{t+1}=N_{t,a}\times S_{a}+N_{t,i}\times R_{i}\times S_{i}}
159:
I = number of individuals immigrating into the population between N
169:
E = number of individuals emigrating from the population between N
355:= annual survival of immature females from time t to time t+1
348:= annual survival of adult females from time t to time t+1
861:
763:
663:
542:
458:
385:
653:
375:
219:
51:
149:
D = number of deaths within the population between N
139:
B = number of births within the population between N
366:In matrix notation this model can be expressed as:
940:
600:
315:
110:
200:methods, then multiplying present abundance and
995:Leslie Matrix Model demonstration (Silverlight)
8:
915:
910:
894:
889:
873:
868:
856:
837:
808:
794:
782:
770:
758:
735:
724:
708:
697:
681:
670:
658:
654:
652:
575:
570:
554:
549:
537:
523:
511:
497:
487:
475:
465:
453:
430:
419:
403:
392:
380:
376:
374:
307:
294:
275:
262:
243:
224:
218:
210:The BIDE model can then be expressed as:
75:
56:
50:
341:= number of immature females at time t
111:{\displaystyle N_{t+1}=N_{t}+B-D+I-E,}
7:
334:= number of adult females at time t
14:
972:Population dynamics of fisheries
27:. Population models are used in
1:
188:, often using some form of
1031:
17:Matrix population models
129:= abundance at time t+1
19:are a specific type of
942:
602:
317:
112:
943:
603:
318:
136:= abundance at time t
113:
651:
373:
217:
49:
1010:Population dynamics
938:
933:
925:
850:
745:
598:
593:
585:
531:
440:
313:
198:mark and recapture
190:mark and recapture
108:
29:population ecology
1015:Population models
1022:
947:
945:
944:
939:
934:
930:
929:
922:
921:
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919:
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536:
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492:
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410:
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322:
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314:
312:
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299:
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286:
285:
267:
266:
254:
253:
235:
234:
117:
115:
114:
109:
80:
79:
67:
66:
21:population model
1030:
1029:
1025:
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1023:
1021:
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1019:
1000:
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968:
961:
957:
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716:
704:
693:
690:
689:
677:
666:
659:
649:
648:
643:
639:
635:
631:
628:. Note, that F
627:
623:
619:
592:
591:
584:
583:
571:
566:
563:
562:
550:
545:
538:
530:
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290:
271:
258:
239:
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176:
172:
166:
162:
156:
152:
146:
142:
135:
128:
71:
52:
47:
46:
42:can be modeled
12:
11:
5:
1028:
1026:
1018:
1017:
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1002:
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998:
997:
992:
979:
976:
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928:
918:
914:
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905:
904:
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876:
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823:
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815:
811:
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789:
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777:
773:
769:
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764:
762:
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748:
738:
734:
730:
727:
723:
719:
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707:
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669:
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629:
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193:
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140:
137:
133:
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119:
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83:
78:
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59:
55:
25:matrix algebra
13:
10:
9:
6:
4:
3:
2:
1027:
1016:
1013:
1011:
1008:
1007:
1005:
996:
993:
990:
989:0-87893-096-5
986:
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981:
977:
973:
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965:
963:
952:
935:
926:
916:
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907:
895:
891:
886:
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865:
858:
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845:
838:
834:
828:
821:
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805:
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783:
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771:
767:
760:
755:
753:
746:
736:
732:
728:
725:
721:
709:
705:
701:
698:
694:
682:
678:
674:
671:
667:
660:
647:
646:
645:
614:
613:Leslie matrix
595:
586:
576:
572:
567:
555:
551:
546:
539:
532:
524:
520:
512:
508:
498:
494:
488:
484:
476:
472:
466:
462:
455:
450:
448:
441:
431:
427:
423:
420:
416:
404:
400:
396:
393:
389:
382:
369:
368:
367:
357:
350:
343:
336:
329:
328:
327:
308:
304:
300:
295:
291:
287:
282:
279:
276:
272:
268:
263:
259:
255:
250:
247:
244:
240:
236:
231:
228:
225:
221:
213:
212:
211:
208:
205:
203:
202:survival rate
199:
191:
181:
168:
158:
148:
138:
131:
124:
123:
122:
105:
102:
99:
96:
93:
90:
87:
84:
81:
76:
72:
68:
63:
60:
57:
53:
45:
44:
43:
41:
36:
34:
31:to model the
30:
26:
22:
18:
953:
950:
610:
365:
325:
209:
206:
182:
179:
120:
37:
16:
15:
954:The terms F
40:populations
1004:Categories
978:References
23:that uses
301:×
288:×
256:×
100:−
88:−
966:See also
33:dynamics
636:×R
326:where:
121:where:
987:
644:term.
958:and S
624:and F
173:and N
163:and N
153:and N
143:and N
985:ISBN
38:All
632:= S
620:, F
339:t,i
332:t,a
175:t+1
165:t+1
155:t+1
145:t+1
127:t+1
1006::
991:.
960:i
956:i
936:.
927:)
917:3
913:t
908:N
896:2
892:t
887:N
875:1
871:t
866:N
859:(
852:)
846:0
839:2
835:S
829:0
822:0
817:0
810:1
806:S
796:3
792:F
784:2
780:F
772:1
768:F
761:(
756:=
747:)
737:3
733:l
729:+
726:t
722:N
710:2
706:l
702:+
699:t
695:N
683:1
679:l
675:+
672:t
668:N
661:(
642:3
638:i
634:i
630:1
626:3
622:2
618:1
616:F
596:.
587:)
577:a
573:t
568:N
556:i
552:t
547:N
540:(
533:)
525:a
521:S
513:i
509:S
499:i
495:R
489:a
485:S
477:i
473:R
467:i
463:S
456:(
451:=
442:)
432:a
428:l
424:+
421:t
417:N
405:i
401:l
397:+
394:t
390:N
383:(
360:i
358:R
353:i
351:S
346:a
344:S
337:N
330:N
309:i
305:S
296:i
292:R
283:i
280:,
277:t
273:N
269:+
264:a
260:S
251:a
248:,
245:t
241:N
237:=
232:1
229:+
226:t
222:N
194:i
186:t
184:N
171:t
161:t
151:t
141:t
134:t
132:N
125:N
106:,
103:E
97:I
94:+
91:D
85:B
82:+
77:t
73:N
69:=
64:1
61:+
58:t
54:N
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