We now discuss populations with discrete breeding seasons, where reproduc- tion is limited to a particluar season of the year. For example, let's consider the number of female ruby-throated hummingbirds in a population with an annual breeding season of March to July. A female usually lays one clutch of two eggs; sometimes two clutches are laid. We will assume that the average life span of a female hummingbird is four years. We define the age of a bird at the END of a breeding season as follows: age zero (0) : any bird that is born during the current breeding season age one (1): any zero-year old bird which survives to the end of the next breeding season age two (2): any one-year old bird which survives to the end of the next breeding season age three (3): any two-year old bird which survives to the end of the next breeding season We make the following assumptions about the reproductive viability of female birds: age zero (0): not yet reproductively mature age one (1): will produce an average of 1.2 female offspring the next breeding season which survive age two (2) will produce an average of 1.5 female offspring the next breeding season which survi age three (3): will produce an average of 0.7 female offspring the next breeding season which survive We make the following assumptions about the survival rates of female birds: 50% of age zero (0) females at time t survive to time t + 1; 35% of age one (1) females at time t survive to time t + 1; 15% of age two (2) females at time t survive to time t + 1; 0% of age three (3) females at time t survive to time t + 1 Be prepared to find and use the "transition matrix" A for a model like this, which satisfies xk+1 = Axk, where the entries in x give the number of birds in each age category at the end of breeding season k.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(6) Population Modeling with Discrete Dynamical Linear Systems.
We now discuss populations with discrete breeding seasons, where reproduc-
tion is limited to a particluar season of the year. For example, let's consider
the number of female ruby-throated hummingbirds in a population with an
annual breeding season of March to July. A female usually lays one clutch of
two eggs; sometimes two clutches are laid. We will assume that the average
life span of a female hummingbird is four years. We define the age of a bird
at the END of a breeding season as follows:
age zero (0) : any bird that is born during the current breeding season
age one (1): any zero-year old bird which survives to the end of the next
breeding season
age two (2): any one-year old bird which survives to the end of the next
breeding season
age three (3): any two-year old bird which survives to the end of the
next breeding season
We make the following assumptions about the reproductive viability of female birds:
age zero (0): not yet reproductively mature
age one (1): will produce an average of 1.2 female offspring the next
breeding season which survive
age two (2) will produce an average of 1.5 female offspring the next
breeding season which survive
age three (3): will produce an average of 0.7 female offspring the next
breeding season which survive
We make the following assumptions about the survival rates of female birds:
50% of age zero (0) females at time t survive to time t +1;
35% of age one (1) females at time t survive to time t + 1
15% of age two (2) females at time t survive to time t +1 ;
:
0% of age three (3) females at time t survive to time t + 1.
Be prepared to find and use the "transition matrix" A for a model like this, which satisfies xk+1 = Axk, where the entries in
*k give the number of birds in each age category at the end of breeding season k.

Expert Solution

Step 1: Writing down the given information

Given the following data:

Assumptions on reproductive viability:

Age zero (0): not yet reproductively mature.
Age one (1): will produce an average of 1.2 female offspring the next breeding season which survive.
Age two (2): will produce an average of 1.5 female offspring the next breeding season which survive.
Age three (3): will produce an average of 0.7 female offspring the next breeding season which survive.

Assumptions on survival rates of female birds:

50% of age zero (0) females at time t survive to time t + 1;
35% of age one (1) females at time t survive to time t + 1;
15% of age two (2) females at time t survive to time t + 1;
0% of age three (3) females at time t survive to time t + 1.