Hi there!
I currently am working on a math assignment where I am required to investigate population trends of the Tasmanian devil since the first documentation of the Devil Facial Tumor Disease (DFTD) in 1996. In order to do this we are required to utilize Leslie Matrices, and of course, when working with Leslie Matrices, the initial population of each age group is required. However, we are not explicitly given the initial age distributions for the Tasmanian devil; rather, we were given the TOTAL population. I have made a start on attempting to work out these initial age group populations, and would like some clarification on the method I have used.
We have been provided with the following data (note that all these figures are for disease-free populations):
Survival Rates (where s0 = probability of surviving the age interval of 0-1 years)
s0=0.39
s1=0.82
s2=0.82
s3=0.82
s4=0.82
s5=0.27
s6=0
Breeding numbers (female per female devil, where m0 = offspring produced by a devil within the age interval of 0-1 years)
m0=0
m1=0.03
m2=0.86
m3=1.55
m4=1.55
m5=1.55
m6=0.86
Total population recorded right before documentation of DFTD disease
130 000.
My solution for finding the initial population per age group:
Let a = the initial number of devils within the age interval of 0-1 in any given year.
Assuming that the given survival rates are constants
Then the initial number of devils within the age interval of 1-2 in any given year must = 0.39a (from s1=0.39).
Similarly...
Initial number of devils within the age interval of 2-3 = 0.39 * 0.82a
= 0.3198a
Initial number of devils within the age interval of 3-4= 0.39 * 0.82 * 0.82a
= 0.262236a
Performing similar calculations above for each respective age group and adding each of the yielding results (i.e. a+0.39a+0.3198a+0.262236a…) gives the following total population in any given year in relation to ‘a’:
Total population = 2.411005427a
Now given that the total population when the disease was documented in 1996 was 130000, age-group populations can be found as follows:
2.411005427a=130000
a=53 919.4
Substituting this value for ‘a’ into previous equations using basic algebra, the following age group populations were given.
With the data I am given, would this be an appropriate method of determining the initial populations?
And one final question. I realize that we are required to model the population decline after the documentation of the disease, and that the survival rates we have been given above are thus not applicable; I have assumed that birth rates will remain the same. We have been provided with the following research that as of now I am speculating we are to use to determine survival rates for the disease-ridden populations, but I am a bit unsure of where to start. Here is a screenshot of the findings:
Since from 1996 to 2007, 50% of the population was killed, would halving the 'healthy' survival rates work to model the decline during this time period?
Apologies for such the long question. Any help would be greatly appreciated.
I currently am working on a math assignment where I am required to investigate population trends of the Tasmanian devil since the first documentation of the Devil Facial Tumor Disease (DFTD) in 1996. In order to do this we are required to utilize Leslie Matrices, and of course, when working with Leslie Matrices, the initial population of each age group is required. However, we are not explicitly given the initial age distributions for the Tasmanian devil; rather, we were given the TOTAL population. I have made a start on attempting to work out these initial age group populations, and would like some clarification on the method I have used.
We have been provided with the following data (note that all these figures are for disease-free populations):
Survival Rates (where s0 = probability of surviving the age interval of 0-1 years)
s0=0.39
s1=0.82
s2=0.82
s3=0.82
s4=0.82
s5=0.27
s6=0
Breeding numbers (female per female devil, where m0 = offspring produced by a devil within the age interval of 0-1 years)
m0=0
m1=0.03
m2=0.86
m3=1.55
m4=1.55
m5=1.55
m6=0.86
Total population recorded right before documentation of DFTD disease
130 000.
My solution for finding the initial population per age group:
Let a = the initial number of devils within the age interval of 0-1 in any given year.
Assuming that the given survival rates are constants
Then the initial number of devils within the age interval of 1-2 in any given year must = 0.39a (from s1=0.39).
Similarly...
Initial number of devils within the age interval of 2-3 = 0.39 * 0.82a
= 0.3198a
Initial number of devils within the age interval of 3-4= 0.39 * 0.82 * 0.82a
= 0.262236a
Performing similar calculations above for each respective age group and adding each of the yielding results (i.e. a+0.39a+0.3198a+0.262236a…) gives the following total population in any given year in relation to ‘a’:
Total population = 2.411005427a
Now given that the total population when the disease was documented in 1996 was 130000, age-group populations can be found as follows:
2.411005427a=130000
a=53 919.4
Substituting this value for ‘a’ into previous equations using basic algebra, the following age group populations were given.
| Age group | 0-1 | 1-2 | 2-3 | 3-4 | 4-5 | 5-6 | 6-7 |
| Initial population | 53 919 | 21 028 | 17 243 | 14 139 | 11 594 | 9507 | 2567 |
With the data I am given, would this be an appropriate method of determining the initial populations?
And one final question. I realize that we are required to model the population decline after the documentation of the disease, and that the survival rates we have been given above are thus not applicable; I have assumed that birth rates will remain the same. We have been provided with the following research that as of now I am speculating we are to use to determine survival rates for the disease-ridden populations, but I am a bit unsure of where to start. Here is a screenshot of the findings:
Since from 1996 to 2007, 50% of the population was killed, would halving the 'healthy' survival rates work to model the decline during this time period?
Apologies for such the long question. Any help would be greatly appreciated.