## Matrix Models

As organisms age, their realized fecundity as well as their survival rates can change. Age-based matrix models separate populations into different age classes and each age class can possess different fecundity and survival rates. Matrix models are useful when hypothesizing about the importance of specific population parameters for certain age classes. Identifying the most sensitive age class is important for management strategies, especially for endangered species.

each age class into a vector n and obtaining the product of the Mn matrices:

Figure 4 Population growth of a model with time lags. The heavy black line was modeled without a time lag (r = 0) and reaches a stable equilibrium point, the light black line was created with a r = 1 and exhibits damped oscillations, and the dotted line was created using r = 2 and exhibits a two-endpoint stable limit cycle.

20 30

Time

Figure 4 Population growth of a model with time lags. The heavy black line was modeled without a time lag (r = 0) and reaches a stable equilibrium point, the light black line was created with a r = 1 and exhibits damped oscillations, and the dotted line was created using r = 2 and exhibits a two-endpoint stable limit cycle.

The most common matrix model is an age-based matrix, commonly referred to as the Leslie matrix. The individuals in a population are separated into classes based on their age (Nx) and each class is assigned a probability of survival (Px) and a realized fecundity (Fx). Fx and Px are put into a transition matrix M with Fx along the top row of the matrix and Px along the subdiagonal, and all other elements in M are assigned a value of zero (Figure 5a). Population projections can be made by placing the current number of individuals in

F0 |
F1 |
F2 .. |
Fm-1 |
Fm |
N0 | |

P0 |
0 |
0 .. |
0 |
0 |
N1 | |

0 |
P1 |
0. |
0 |
0 |
Nm- | |

0 |
0 |
P2 . |
0 |
0 |
nA = | |

0 |
0 |
0. |
Pm-1 |
0 |
G2 P2 0 F3 0 G3 N2 N3 Figure 5 Population projection matrices. (a) An age-structured Leslie matrix (Ma) and a vector (nA of the number of the individuals in each age-class. (b) A stage-based matrix (MB) and a vector (nB) of the number of the individuals in each age class. Subscripts represent the age or size class with m being the oldest or largest class, respectively. where nt is the vector whose elements are the number of individuals in each age class, and nt +1 is the vector containing the number of individuals in each age class at time t + 1. For some species, size or stage of development may be better indicators of an individual's fecundity and survival rate. These models do not assume that every individual leaves the class at every time step. The probability an individual will survive and remain in class x during the next time step is Gx. Stage-based matrices incorporate this term along the diagonal in the transition matrix M. Px is defined as the probability that an individual will move into the succeeding class x + 1. The Px elements are placed along the subdiagonal in the transition matrix M (Figure 5b). Fx is defined the same as in the Leslie matrix and population projections are calculated the same as above. The standard stage-based matrix model assumes that the first size class is not reproductively mature and any the remaining classes each have a specific realized fecundity. |

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