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QUESTION 2 The Leslie matrix for a certain endangered species is given. 0.8 24 03 L=04 0 0 0 055 O A group of the species was moved into a secure property at the start of 2018. The initial female population is given. 150 N() = 80 40 The best estimate of the total female population at the start of 2025 is (A) 3000 (B) 4000 (C) 5000 (D) 6000
QUESTION 11 (5 marks) Teams A B C D and E participated in a competition with the following results: » A defeated D. * B defeated A C and E. e C defeated A and E. e D defeated B C and E. » E defeated A. To rank the teams at the end of the competition the organisers constructed a dominance matrix N that is partially completed. a) By allocating 1 to represent ‘defeated’ and 0 to represent either ‘was defeated by’ or ‘no result’ complete matrix N. [1 mark] Losing teams ooooooooooooooooooooooooooooooooooo ----- A » B g | heeiihociihociiociiond B8l ieaes Siwisss EEEesie) GHESEES wseitese a2 N= cuC .......... | Feeeet et Bevene ool Tuaeed g ooooooooooooooooooooooooooooooooooo = D [: i i B ................................... E The organisers need to rank the teams into individual places from first to fifth place. They decide to use the ranking model N + N? to achieve this. b) Use the model N + N? to rank the teams. [2 marks] c) Use the result from 11b) to identify a limitation of the organisers’ ranking model. [1 mark] d) State a mathematical refinement the organisers could consider to overcome the limitation of the ranking model identified in 11c). [1 mark]
QUESTION 14 (5 marks) The Tasmanian thornbill is a species of bird that has an average life span of three years. Female thornbills do not reproduce in their first year but produce an average of four female offspring in each of their second and third years. The survival rate of each age group is estimated as 25% in their first year and 30% in their second year. A Leslie matrix L modelling the population distribution of the Tasmanian thornbill has been partially completed. = O s S O B a) State the values of x and y. [1 mark] At the start of 2021 a study began into the population of Tasmanian thornbills. The study: * estimated that the initial female population was 510 in their first year 480 in their second year and 420 in their third year found that the ratio of male to female was approximately 1:2. b) Estimate the total population of Tasmanian thornbills at the start of 2025. [4 marks]
QUESTION 2 The win/draw/loss results after a netball competition involving five teams is represented in matrix M. = I Winning teams ~ O w2 T 0 Losing teams Q 1 1 R 2 2 S 0 0 0 Key: Team P drew with Team Q defeated Team R and Team T and lost to Team S The model M+M?+M? is used to rank the teams. The final positions from first to fifth are (A) S Q PR T B) S Q PT R © S P QT R D) S P Q R T
QUESTION 12 (5 marks) A scientist collects data for a species of tree frog in a protected area. Details for the female tree frog population are shown in the table. Swvame | 00 | 0 | 02 | 0 The scientist uses a Leslie matrix model to make predictions about the female tree frog population. a) State the initial population matrix. [1 mark] b) Determine the Leslie matrix. [1 mark] A species 1s considered to be endangered if the female population in a restricted area is predicted to fall to less than 125 in the next 20 years. c) Determine whether this species of tree frog is considered to be endangered. [3 marks]
QUESTION 9 The win/loss results after a soccer competition involving four teams are represented in the matrix M shown below. Losing teams A B C D o A 0 2 1 0 = M = § B 0 0 0 | EJ 2 C ] 2 0 | 5 = D 2 ] 1 0 Key: Team A defeated Team B drew with Team C and lost to Team D. Given the model M + M? + M3 to rank the teams determine their final positions from first to fourth. (A) C A D B (B) C D A B (C) D A C B (D) D C A B
QUESTION 15 (6 marks) A research student has collected fertility and survival rates for a certain endangered species over a number of years. A small group of this species has been moved into a secure property. The Leslie matrix for the survival rates of the species is 0 0.1 0.8 0.5 095 O 0 0 0 083 0 0 0 0 064 0 The female population of the species moved into the property at the start of Year 1 is 5 & 50 tl2s 20 a) Explain the meaning of the number 0.8 in the Leslie matrix. [1 mark] b) Calculate the expected total female population in the property at the start of Year 2. [2 marks] c) Determine the expected total population in the property at the start of Year 11 based on 55% of the population being female. [3 marks]
QUESTION 4 The age-specific population distribution of a particular species of animal is shown. see eae) redngrae | 0) Sunivalewe | 06 | os) oe The Leslie matrix based on this data is 94 82 37 6 (Ay [06 0 0 0 0 08 0 0 L0 0 04 0 1 2 3 4 (B) |13 0 0 0 09 0 0 0 0 02 0 0 09 0 oo Oo O&O a (D) 0 08 0 0 0.6 0.8 0.4 (C) 13 0 O 0 1.3 0.9 0.2 06 0 O 0
QUESTION 7 Matrix N represents the results for a competition involving four teams. Losing teams P QR S_ a P 0 01 1 Nae wo 2/1 0 0 0 Ss =RiO 1 0 0 & = S10 1 1 0. Key: Team P lost to team Q but won against teams R and S. Using the ranking model N + 0.5N7 the teams that placed first second and third respectively are (A) PS andQ. (B) P SandR. (C) S PandQ. (D) S PandR.
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