Question

Q2–Lefkovitch Matrices and Population of Aedes Aegypti

Lefkovitch Matrices are similar to Leslie Matrices, except they allow individuals to stay in the same stage OR move on. We’ll use them to study a population of mosquitoes called Aedes Aegypti. Let’s make six assumptions about Aedes’ behaviour (all instances of “eggs” in the following only refer to female eggs).

1. There are 4 stages: egg, baby (larva), youth (pupa), and adults.

2. On average, female youths produce 2 eggs a week and female adults produce 3 eggs a

   week. Babies do not produce eggs.

3. 70% of eggs hatch every week. Among the remaining eggs, half are eaten by predators, while half stay as eggs.

4. 20% of babies are captured (and thus removed) every week, while 50% become youth.

5. 50% of youth are captured every week, while 20% become adults.

6. 60% of adults are captured every week.

Let Ei be the number of female eggs, Bi be the number of female babies, Yi be the number of

female youth, and Ai be the number of female adults at the ith week. Questions:

(a) Write down equations for each of Ei+1, Bi+1, Yi+1, and Ai+1 in terms of Ei, Bi, Yi, and Ai. Hint: Try to lay out as much information as possible in a Flow Diagram.

(b) Find matrix L so that

E: Bi L = Ei+1 Bi+1 Yi+1 Air1. Y; Ai

(c) In a field of tall grass, there are initially no eggs, no babies, 100 female youths and 50 female adults. In two weeks from now, what will the population distribution be?

Answer 1

Let, E : The no g Fornale eggs at the the week Bi : The hoog female babies 1/ 11 11 4 11 и Y; : The no. of female you the Aj : The no- of female adults 4 1 11 using the information given: Egns for Erei, Bity Yitia fit in terms of Ei, Bi, Yigai flow diagram eaten by 159. Bosphore Logo captured of captured bot Predators יע Bi in the ith week a Y; Ai {(301) 70 yoy. 204. 407 2 15t. +367 3ot In the citith week : Gill Biel Hier Ail 30/1 30 2 0.30 Mence, the egas are! Eith (0.15) Ei + (O) Bi 7 (2.14; +(3) A Bill :79) B 1 (0 : 0) Bit O), (-) h) 1 o) E Yies = (Dobsontoret + 0.50) Bi + (0:30) Y; toa Hill Col5 t (0) + (0-20) 4; +(0.49)

rusting eno mappy from we we get o.15 0 2 3 Gel G 0:30 o Biel 0.70 Yi1 0.30 Yi Ara o 0:40 AI 0.15 2 3 Giel OR bi 0.70 Bill 0:30. 0 Bi' o oso 030 Y; Yin 0.40 O: 20 Which is of the form GT Giul L Bi Bien Y; A Aral to Hence, The mabix L (of order 4x4) his 0:15 2 3 L2 0.70 0:30 o Ono 0.30 o 0 0:40 ido 0: 00.20 © In In a freld a tall grass, 19 izo ta week Initially, ie o tit Even Bo 2 Yo 2 to A. so

In 1 week from Now Tiel 5 oils 2 3 0-70 B, 0.30 o Bo Oyo 0:30 o Yo A 0:20 o.15 2 3 = 0.70 d. 30 O O O'bo 0.30 lao to O 0.20 0:40 Ei 3 5 350, = ) B, B, 2 Y 30 Lai, Y, > 30 40 =40 of two weeks bram Now / 0:15 2 · 3 50 wa 0.70 0:30 0-30 30 Y 0-120 0.40 40 AL Ez 232-5 Ę2 23 2.5 ~ 232 -) B₂ 245 =) B₂ 2245 Y, 2 9 22 A2 222 i. In two weeks from woll be now the The population disby botan B = 245 Y₂ 29 & Az = 22 Ę 2 2 32.5 ~ 232 Ang /