Ecological Archives E087-141-A1

Shana K. Mertens, Jonathan M. Yearsley, Frank van den Bosch, and Christopher A. Gilligan. 2006. Transient population dynamics in periodic matrix models: methodology and effects of cyclic permutations. Ecology 87:2338–2348.

Appendix A. Effects of cyclic permutations and initial conditions on the transient dynamics of a tropical grass (Andropogon semiberbis) exposed to alternating burn and non-burn environments.

In this example we use a model of the perennial savanna grass Andropogon semiberbis (Silva et al. 1991). The life-history of A. semiberbis is strongly affected by the action of fire, leading to two environments, burned (B) and unburned (U), represented by yearly transition matrices B and U, respectively. The population is structured by the number of tillers per plant, giving four size classes: 1 tiller (Class 1), 2–10 tillers (Class 2), 11–20 tillers (Class 3) and >20 tillers (Class 4). The life-cycle graph, based upon these classes, is shown in Fig. A1 and the rates for the life-history transitions are in Table A1. The effect of burning is to increase the life-history transition rates, and to create a couple of new transitions which are not seen in an unburned environment (Silva et al. 1991).


Life cycle of <i>Andropogon semiberbis</i>
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   FIG. A1. The life-cycle of Andropogon semiberbis (Silva et al. 1991). The dotted lines indicate transitions that occur in the burn phase only. Table A1 gives the values associated with each life history transition for both phases. The numbers indicate the various size classes.

The long-term population growth rates over one cycle are the same for these two permutations, at 0.42~per two-year cycle. The long-term periodic sensitivities (Table A1) and elasticities are also independent of the cyclic permutation of the environments (Caswell 2001). In the case of the permutations BU and UB, perturbations to b21 and b22 will have the largest effect in the burn phase, while perturbations to u12 and u22 are most important in the unburn phase (Table A1). Likewise, the average long-term stable structure is identical for all cyclic permutations of the environments, but the stage structure does change within a cycle. After a B year less than 60% of the population is in the first size class, whilst after a U year more than 75% of the population is in the first size class (Table A1).

We first examine how the cyclic permutations affect transient population dynamics and consider cyclic permutations UB and BU. We use a total initial population size of n(0) = 1000, with an initial structure where the entire population is concentrated in the smallest size class (Class 1). We then examine how different initial conditions interact with cyclic permutations. We again use a total initial population size of n(0) = 1000 and consider all extreme initial population structures (i.e., the population is concentrated in a single stage class). We present our results in terms of time-scaled elasticities (Fox and Gurevitch 2000), which can be compared with the elasticities of the long-term population growth rate.


TABLE A1. Life-history transition matrices for Andropogon semiberbis growing in burn (B) and unburn (U) environments (Silva et al. 1991), and the elasticity values and stable size structure resulting from asymptotic analysis of the cyclic permutations of BU and UB. In the matrices, the element at row 1, column 2 quantifies the transition from class 2 to class 1. The asymptotic growth rate associated with the permutations BU and UB is λ1 = 0.42 per two-year cycle.


Phase  
Transition matrix
 
Elasticity (permutation BU and UB)
 
Stable structure

 
Class
1
2
3
4
 
1
2
3
4
 
Burn (B)
1
0.080
1.630
2.420
4.400
 
0.006
0.024
0.017
0.002
 
0.584
 
2
0.210
0.640
0.350
0.160
 
0.361
0.228
0.062
0.002
 
0.317
 
3
0
0.190
0.430
0.240
 
0
0.054
0.061
0.002
 
0.071
 
4
0
0.030
0.230
0.480
 
0
0.033
0.127
0.020
 
0.028
                         
Unburn (U)
1
0
0.706
0.391
3.590
 
0
0.233
0.029
0.105
 
0.759
 
2
0.018
0.158
0.136
0.093
 
0.049
0.233
0.045
0.012
 
0.157
 
3
0
0.080
0.070
0.210
 
0
0.187
0.037
0.043
 
0.078
 
4
0
0
0.010
0.070
 
0
0
0.007
0.019
 
0.006


 

Effects of cyclic permutations

The cyclic permutations unburn-burn (UB) and burn-unburn (BU) create different transient population dynamics which ultimately produce a permanent effect on population size (Fig. A2). When the population is initially concentrated in the smallest population class (Class 1), the UB permutation has a population size that is an order of magnitude lower than in the BU permutation (Fig. A2). When the population is initially concentrated in the smallest size class (Class 1), starting with a U phase reduces the population much more than a B phase, because all the transition rates from Class 1 are weaker in the U environment compared to the B environment (Table A1).


Effects on population size
 
   FIG. A2. The effects of initial population structure and cyclic permutation on population size, at the end of each phase of a two-phase cycle. Each line shows the population dynamics for a different combination of initial population structure and cyclic permutation. The total initial population size is N(0) = 1000 and the initial stage structures are all 1000 individuals concentrated in either size class 1, 2, 3, or 4.

The order of the environments affects both the pattern of elasticities and the persistence of the transient period Fig. A3. The immediate elasticities for both the B and U environments differ between cyclic permutations (compare Fig. A3a and b, and Fig. A3d and e, respectively). With regard to the B environment, the ranking of elasticities is substantially different between the two permutations, as well as the persistence of the transient dynamics. In the BU cycle, the immediate elasticity associated with b21 begins at 0.72 and then decreases to approach its asymptotic value of 0.36 (Fig. A3a). The elasticities associated with b22 and b43 begin close to zero and gradually increase to their asymptotic values of 0.23 and 0.13, respectively. On the other hand, in the UB cycle (Fig. A3b), the highest immediate elasticity is initially to b12 (immediate elasticity = ~0.65), and then elements b22 (cycles 2 to 7) and b21 (cycle 7 onwards) are successively ranked highest. The elasticity to b22 increases sharply between the first and second cycles and then gradually decreases to its asymptote at 0.23, whereas b21 begins close to zero and increases to its asymptotic value of 0.36. When one examines the immediate elasticities of the U phase (Fig. A3d and e), it is clear that persistence of the transient period can differ substantially between cyclic permutations. In the UB cycle (Fig. A3e), the elements u22 and u32 take much longer to reach their asymptotic elasticities (0.23 and 0.19, respectively) than in the BU cycle (Fig. A3d).

The different patterns of elasticities can be understood by considering the initial population structure and its subsequent time evolution. For example consider the element b21 and the initial conditions n(0) = (1000, 0, 0, 0)T. In cycle UB, a perturbation of this element will initially have no effect on population size in contrast to cycle BU (Fig. A3a and b). This occurs because in the UB cycle, at the start of the B phase, the entire population is concentrated in Class 2, as the only transition the population has experienced is u21 (Table A1). Consequently in the B phase, only transitions which move the population from Class 2, will affect population size. Therefore, in the UB cycle, the immediate elasticity associated with b21 is zero and then increases towards 0.36 as the population in Class 1 increases to reach its stable stage structure. The high initial value the immediate elasticity associated with b21 (e21 = ~0.75) in the BU cycle can be explained using similar lines of reasoning.


Effects of cyclic permutations on transient elasticity
 
   FIG. A3. Immediate and end-of-cycle transient elasticities of population size scaled for time, for the two cyclic permutations BU and UB. The initial population has 1000 individuals concentrated in the smallest size class (Class 1). Results are shown only for elements whose transient elasticities are about 0.1 or greater at some time. Panels a–c are the results for the burn environment, and panels d–f are for the unburn environment.

Initial population stage structure

The initial population structure has a long-term effect on the total population size of over an order in magnitude (Fig. A2). Relatively large population sizes result from an initial structure which concentrates individuals in Class 4, whilst relatively small population sizes emerge from an initial structure which concentrates individuals in Class 1. These differences emerge from the transient dynamics in the first couple of years, after which all populations approach the same long-term growth rate.

The initial population structure can also have a large effect on the elasticities of the population size. For the immediate elasticity associated with element b21, all combinations of initial conditions with cyclic permutations result in a different approach to the asymptotic value (Fig. A4). In the UB cycle, when the initial population is concentrated in Class 4, the immediate elasticity of population size to perturbations b21 quickly reaches its asymptotic value. In contrast, when the initial population 2 is concentrated in Class 1, then the elasticity associated with b21 begins at zero and slowly increases to reach its asymptotic value.


Effects of ic on transient elasticity
 
   FIG. A4. The effect of different initial stage structures on the time-scaled elasticity (elasticity divided by number of cycles) of population size to perturbations to the transition from the first to the second size class, in the burn environment, b21 = 0.21. The asymptotic elasticity is e21 = 0.361 (Table A1).

LITERATURE CITED

Caswell, H. 2001. Matrix population models: construction, analysis, and interpretation. Second edition. Sinauer Associates, Sunderland, Massachusetts, USA.

Fox, G. A., and J. Gurevitch 2000. Population numbers count: Tools for near-term demographic analysis. American Naturalist 156:242–256.

Silva, J. F., J. Raventos, H. Caswell, and M.C. Trevisan. 1991. Population responses to fire in a tropical savanna grass, Andropogon semiberbis: A matrix model approach. Journal of Ecology 79:345–356.



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