896:
442:
891:{\displaystyle {\begin{bmatrix}n_{0}\\n_{1}\\\vdots \\n_{\omega -1}\\\end{bmatrix}}_{t+1}={\begin{bmatrix}f_{0}&f_{1}&f_{2}&\ldots &f_{\omega -2}&f_{\omega -1}\\s_{0}&0&0&\ldots &0&0\\0&s_{1}&0&\ldots &0&0\\0&0&s_{2}&\ldots &0&0\\\vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&0&\ldots &s_{\omega -2}&0\end{bmatrix}}{\begin{bmatrix}n_{0}\\n_{1}\\\vdots \\n_{\omega -1}\end{bmatrix}}_{t}}
1317:. Then the non-trivial, effective eigenvalue which defines the long-term asymptotic dynamics of the mean-value population state vector can be presented as the effective growth rate. This eigenvalue and the associated mean-value invariant state vector can be calculated from the smallest positive root of a secular polynomial and the residue of the mean-valued Green function. Exact and perturbative results can thusly be analyzed for several models of disorder.
1300:
provides a means of identifying the intrinsic growth rate. The stable age-structure is determined both by the growth rate and the survival function (i.e. the Leslie matrix). For example, a population with a large intrinsic growth rate will have a disproportionately “young” age-structure. A population
1309:
There is a generalization of the population growth rate to when a Leslie matrix has random elements which may be correlated. When characterizing the disorder, or uncertainties, in vital parameters; a perturbative formalism has to be used to deal with linear non-negative
1152:
provides the stable age distribution, the proportion of individuals of each age within the population, which remains constant at this point of asymptotic growth barring changes to vital rates. Once the stable age distribution has been reached, a population undergoes
43:(also called the Leslie model) is one of the most well-known ways to describe the growth of populations (and their projected age distribution), in which a population is closed to migration, growing in an unlimited environment, and where only one sex, usually the
1042:
980:
65:
The Leslie matrix is a square matrix with the same number of rows and columns as the population vector has elements. The (i,j)th cell in the matrix indicates how many individuals will be in the age class
54:
to model the changes in a population of organisms over a period of time. In a Leslie model, the population is divided into groups based on age classes. A similar model which replaces age classes with
309:
434:
1074:
58:
is called a
Lefkovitch matrix, whereby individuals can both remain in the same stage class or move on to the next one. At each time step, the population is represented by a
1239:
1395:
Further details on the rate and form of convergence to the stable age-structure are provided in
Charlesworth, B. (1980) Evolution in age-structured population. Cambridge.
1126:
1100:
1175:
1146:
919:
250:
1294:
378:
339:
213:
178:
141:
105:
1259:
36:
1296:, or age distribution, the population tends asymptotically to this age-structure and growth rate. It also returns to this state following perturbation. The
1407:
M.O. Caceres and I. Caceres-Saez, Random Leslie matrices in population dynamics, J. Math. Biol. (2011) 63:519–556 DOI 10.1007/s00285-010-0378-0
1269:
This age-structured growth model suggests a steady-state, or stable, age-structure and growth rate. Regardless of the initial population size,
74:. At each time step, the population vector is multiplied by the Leslie matrix to generate the population vector for the subsequent time step.
991:
930:
1380:
1458:
1430:
62:
with an element for each age class where each element indicates the number of individuals currently in that class.
1148:, gives the population's asymptotic growth rate (growth rate at the stable age distribution). The corresponding
345:
is simply the sum of all offspring born from the previous time step and that the organisms surviving to time
1482:
255:
383:
1396:
1314:
1050:
1181:
1449:
Pollard, J. H. (1973). "The deterministic theory of H. Bernardelli, P. H. Leslie and E. G. Lewis".
1297:
1198:
1185:
1361:
Hal
Caswell (2001). Matrix Population Models: Construction, Analysis, and Interpretation. Sinauer.
1109:
1083:
1160:
1131:
1301:
with high mortality rates at all ages (i.e. low survival) will have a similar age-structure.
904:
222:
219:. More precisely, it can be viewed as the number of offspring produced at the next age class
1348:
Leslie, P.H. (1948) "Some further notes on the use of matrices in population mathematics".
1272:
356:
317:
191:
156:
119:
83:
40:
8:
1477:
24:
1244:
1154:
59:
32:
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1454:
1426:
1376:
55:
28:
77:
To build a matrix, the following information must be known from the population:
1471:
1333:
Leslie, P.H. (1945) "The use of matrices in certain population mathematics".
1311:
1192:
1373:
Conservation of wildlife populations: demography, genetics, and management
1261:, while the Leslie model may have these sums greater or less than 1.
1149:
1335:
1103:
252:
weighted by the probability of reaching the next age class. Therefore,
185:
181:
1037:{\displaystyle \mathbf {n} _{t}=\mathbf {L} ^{t}\mathbf {n} _{0}}
51:
1442:
Ecology: the experimental analysis of distribution and abundance
1195:. The main difference is that in a Markov model, one would have
975:{\displaystyle \mathbf {n} _{t+1}=\mathbf {L} \mathbf {n} _{t}}
44:
143:, the fraction of individuals that survives from age class
1451:
Mathematical models for the growth of human populations
820:
534:
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994:
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359:
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258:
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194:
159:
122:
86:
1191:
The Leslie model is very similar to a discrete-time
436:. This implies the following matrix representation:
70:at the next time step for each individual in stage
1288:
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1233:
1169:
1140:
1120:
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1068:
1036:
974:
913:
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428:
372:
333:
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244:
207:
172:
135:
99:
1444:(5th ed.). San Francisco: Benjamin Cummings.
921:is the maximum age attainable in the population.
1469:
1389:
1453:. Cambridge University Press. pp. 37–59.
188:average number of female offspring reaching
1425:. Cambridge: Cambridge University Press.
16:Age-structured model of population growth
1448:
1264:
1470:
1375:. John Wiley & Sons. p. 104.
1304:
1439:
1370:
1420:
1102:is the Leslie matrix. The dominant
304:{\displaystyle f_{x}=s_{x}b_{x+1}.}
13:
1414:
429:{\displaystyle n_{x+1}=s_{x}n_{x}}
215:born from mother of the age class
14:
1499:
1076:is the population vector at time
1423:Elements of Mathematical Ecology
1114:
1088:
1069:{\displaystyle \mathbf {n} _{t}}
1056:
1024:
1012:
997:
962:
956:
936:
1352:, 35(3–4), 213–245.
1401:
1364:
1355:
1342:
1327:
1184:of the matrix is given by the
1:
1320:
1234:{\displaystyle f_{x}+s_{x}=1}
50:The Leslie matrix is used in
1121:{\displaystyle \mathbf {L} }
1095:{\displaystyle \mathbf {L} }
107:, the count of individuals (
7:
314:From the observations that
10:
1504:
1397:Cambridge University Press
1298:Euler–Lotka equation
349:are the organisms at time
1371:Mills, L. Scott. (2012).
1182:characteristic polynomial
353:surviving at probability
1170:{\displaystyle \lambda }
1141:{\displaystyle \lambda }
924:This can be written as:
31:that is very popular in
1339:, 33(3), 183–212.
914:{\displaystyle \omega }
245:{\displaystyle b_{x+1}}
1290:
1255:
1235:
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1142:
1122:
1096:
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1038:
976:
915:
892:
430:
374:
335:
305:
246:
209:
174:
137:
101:
1440:Krebs, C. J. (2001).
1291:
1289:{\displaystyle N_{0}}
1256:
1236:
1172:
1143:
1123:
1097:
1071:
1039:
977:
916:
893:
431:
375:
373:{\displaystyle s_{x}}
336:
334:{\displaystyle n_{0}}
306:
247:
210:
208:{\displaystyle n_{0}}
175:
173:{\displaystyle f_{x}}
138:
136:{\displaystyle s_{x}}
102:
100:{\displaystyle n_{x}}
1315:difference equations
1273:
1265:Stable age structure
1245:
1199:
1186:Euler–Lotka equation
1161:
1132:
1110:
1084:
1051:
992:
931:
905:
443:
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111:) of each age class
84:
1305:Random Leslie model
1483:Population ecology
1286:
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1167:
1155:exponential growth
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133:
97:
56:ontogenetic stages
33:population ecology
1382:978-0-470-67150-4
1254:{\displaystyle x}
47:, is considered.
37:Patrick H. Leslie
29:population growth
1495:
1464:
1445:
1436:
1421:Kot, M. (2001).
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1415:Further reading
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23:is a discrete,
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25:age-structured
15:
9:
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4:
3:
2:
1500:
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1475:
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1462:
1460:0-521-20111-X
1456:
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1447:
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1438:
1434:
1432:0-521-00150-1
1428:
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1367:
1358:
1351:
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1326:
1318:
1316:
1313:
1312:random matrix
1302:
1299:
1281:
1277:
1262:
1248:
1228:
1225:
1220:
1216:
1212:
1207:
1203:
1194:
1189:
1187:
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1178:
1164:
1156:
1151:
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1105:
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1002:
988:
987:
986:
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947:
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941:
927:
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925:
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883:
877:
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748:
743:
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628:
621:
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586:
583:
579:
573:
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562:
554:
550:
542:
538:
531:
526:
521:
518:
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498:
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491:
483:
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470:
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438:
437:
421:
417:
411:
407:
403:
398:
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392:
388:
365:
361:
352:
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344:
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260:
237:
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200:
196:
187:
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165:
161:
153:
150:
147:to age class
146:
128:
124:
116:
114:
110:
92:
88:
80:
79:
78:
75:
73:
69:
63:
61:
57:
53:
48:
46:
42:
39:. The Leslie
38:
34:
30:
26:
22:
21:Leslie matrix
1450:
1441:
1422:
1403:
1391:
1372:
1366:
1357:
1349:
1344:
1334:
1329:
1308:
1268:
1193:Markov chain
1190:
1179:
1077:
1046:
984:
923:
900:
350:
346:
342:
313:
216:
148:
144:
112:
108:
76:
71:
67:
64:
49:
35:named after
20:
18:
1150:eigenvector
380:, one gets
1478:Population
1472:Categories
1350:Biometrika
1336:Biometrika
1321:References
1128:, denoted
1104:eigenvalue
186:per capita
1241:for each
1165:λ
1136:λ
909:ω
867:−
864:ω
852:⋮
794:−
791:ω
781:…
759:⋮
754:⋮
749:⋱
744:⋮
739:⋮
734:⋮
717:…
678:…
639:…
605:−
602:ω
587:−
584:ω
574:…
499:−
496:ω
484:⋮
182:fecundity
27:model of
1488:Matrices
1157:at rate
341:at time
52:ecology
1457:
1429:
1379:
1047:where
901:where
184:, the
60:vector
45:female
41:matrix
1455:ISBN
1427:ISBN
1377:ISBN
1180:The
1080:and
985:or:
19:The
1177:.
1106:of
347:t+1
343:t+1
149:x+1
1474::
1188:.
180:,
1463:.
1435:.
1385:.
1282:0
1278:N
1249:x
1229:1
1226:=
1221:x
1217:s
1213:+
1208:x
1204:f
1115:L
1089:L
1078:t
1062:t
1057:n
1030:0
1025:n
1018:t
1013:L
1008:=
1003:t
998:n
968:t
963:n
957:L
953:=
948:1
945:+
942:t
937:n
884:t
878:]
870:1
860:n
843:1
839:n
829:0
825:n
818:[
810:]
804:0
797:2
787:s
776:0
771:0
766:0
727:0
722:0
710:2
706:s
700:0
695:0
688:0
683:0
673:0
666:1
662:s
656:0
649:0
644:0
634:0
629:0
622:0
618:s
608:1
598:f
590:2
580:f
567:2
563:f
555:1
551:f
543:0
539:f
532:[
527:=
522:1
519:+
516:t
510:]
502:1
492:n
475:1
471:n
461:0
457:n
450:[
422:x
418:n
412:x
408:s
404:=
399:1
396:+
393:x
389:n
366:x
362:s
351:t
327:0
323:n
299:.
294:1
291:+
288:x
284:b
278:x
274:s
270:=
265:x
261:f
238:1
235:+
232:x
228:b
217:x
201:0
197:n
166:x
162:f
151:,
145:x
129:x
125:s
113:x
109:n
93:x
89:n
72:j
68:i
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