Applied Linear Algebra: Leslie Matrices

Applied Linear Algebra: Leslie Matrices and The Harvesting of Animal Populations Sabrina Tobe, Collin Hendershot, Margaret McGuire 22 November 2016

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Introduction

In this project we investigate one method of modeling animal harvesting known as the Leslie Matrix Method. A Leslie Matrix is an age-structured model of population growth that is very popular in population ecology. In the Leslie Matrix Method we categorize a given animal population according to age groups and remove a certain fraction of each age group at harvest time. The Leslie Matrix Model makes many assumptions about the population being studied.

In the following list we summarize some of the assumptions of the model and state the harvesting practices that will be used in forthcoming computational problems involving Leslie Matrices: • Sustainable Harvest Policy. We assume the yield of each harvest is the same, and consequently the age distribution of the population after each harvest is the same(Anton).

• Working with Females Only. We limit our discussion to females of the population, but we assume the number of males is equal to the number of females in each age class so that by symmetry our discussion here also applies to the male population(Anton).

• Harvesting Period<<Growth Period. We assume that the harvesting period is so much shorter than the growth period that no growth occurs in the population during the harvesting period(Anton).

• No Immigration or Emigration. Increases or decreases in population size are limited to internal population changes (reproduction, death, harvesting) and do not take into account immigration from or emigration to other populations(Anton).

• Unlimited Environment. The Leslie Matrix assumes that the conditions that allow for the growth experienced in each growth period are the same each period, so there are essentially unlimited resources for the population(Anton).

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Historical Background

Traditionally the Leslie Matrix is attributed to Patrick H. Leslie and the first known appearance of the Leslie Matrix derivation in the literature appears in Biometrika in 1945. Interestingly, two other individuals, Bernardelli (1941) and Lewis (1942) came up up with matrix-based population analysis tools almost exactly the same as that of Leslie with nearly identical assumptions and uses (Quinn). It is unclear why there were so many individuals working independently toward this tool in the forties, or why Leslie was recognized as the sole creator despite developers before him. It may simply be that Leslie’s paper is the most accessible and provides the most rigorous presentation of the derivation and use of the matrix model.

Because of the assumptions inherent to the matrix model, Leslie and his predecessors recognized that the model would be most useful for high density mammalian populations such as sheep or rodents. Leslie himself uses an example involving the brown rat, Rattus norvegicus. Ecologists had already assembled a life table and fertility table for the brown rat that Leslie was able to use to develop a Leslie Matrix for the population. Similarly, in Elementary Linear Algebra Anton references an example concerning sheep populations in New Zealand, where ecologists had extensively studied the fertility and survival rates of sheep populations. In the following section we will see how this ecological data regarding fertility and survival allow us to construct a Leslie Matrix.

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Assembling and Using a Leslie Matrix

To construct a Leslie Matrix, we must collect demographic data about the population. We must know the fecundity, or the per capita average of female offspring reaching x0 born from mother of the age class n. The first row of the Leslie Matrix then consists of these age-specific fertilities as represented by Fn (Jones).

The majority of the rest of the matrix is relatively simple as all entries but a sub-diagonal are 0. This particular sub-diagonal is comprised of age-specific survival probabilities from age class n to age class n + 1 represented by Sn (Jones). Note that the subscript s refers to the maximum age a member of the population may reach. As such, the last entry of Sn is Ss−1 as there is no probability of survival after the final age class (Michigan State University).

A Leslie Matrix constructed this way resembles the following matrix:   F0 F1 F2 · · · Fs S0 0 0 ··· 0    0 S1 0 · · · 0   L=0 0 S2 · · · 0    .. .. .. .. ..   . . . . . 0 0 0 Ss−1 0 We may then be able to estimate a future population vector using the product of a prior population vector divided into age group and the Leslie Matrix 2

(Jones). The population vector may be constructed as such: x(t) = hx1 , x2 , ..., xn i ¯ In which each element of the vector x(t) is the number of individuals in that age-class at time t. Then the product Lx(t) will produce a column vector modelling the future population after a single growth period. Thus vector x(t+1) = Lx(t) (Jones).

This may look familiar as we have the dot product of a matrix and vector, resulting in a second vector, and indeed it is! We have seen this before when solving for eigenvalues. It would turn out that the eigenvalue, λ, which satisfies the equation Lx(t) = λx(t) is crucial to the solution to a stable state for a population(Modeling Population Growth). As per usual λ is calculated using the method of det(L − λI) = 0.

Yet, the use of eigenvalues with population growth models may be difficult to recognize automatically. The answer to this is that each Leslie Matrix has a single dominant eigenvalue λ, or the largest eigenvalue that is unique, real and positive that determines stability; this eigenvalue is always a positive number and ultimately determines the long-term behavior of the model(Kiser). That said, if 0 < λ < 1, then the population eventually decreases to zero. Or in another case, if the dominant eigenvalues λ = 1, the the population approaches a stable steady state. Finally, should the dominant eigenvalue λ > 1, then the population grows boundlessly(Modeling Population Growth).

While understanding how growth and stability of a population is determined by a Leslie Matrix is imperative, it is also important to note that there are a variety of other ways in which we use Leslie Matrices as we will discuss in the following section.

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Mechanics of The Leslie Matrix Model

Using the Leslie Matrix we can model the number of animals in each age group as a vector x(t) = hx1 , x2 , ..., xn i ¯ where each entry of the vector is the number of female animals in a given age group (can also be expressed as a column vector. Then we define the Leslie Matrix (denoted by an upper case L) as a matrix of numbers that describes the growth of a population over the course of a growth period. Thus the product Lx(t) is the age distribution vector of the population at the end of the growth period but before the harvest. We can also define a diagonal harvest matrix

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(denoted by a capital H) of the form:  h1 0  0 h2   .. .. . .

0

0

0 0 .. .

... ... .. .

0 0 .. .

0

...

hn

    

Where fi for i = 1, 2, ..., n are the fraction of females from the ith class that is harvested. From all of this we then know the product   h1 (Lx(t))1  h2 (Lx(t))2    HLx(t) =   ..   . hn (Lx(t))n is the vector containing the number of females harvested from the each class.

Furthermore, since we are assuming a sustainable harvesting policy, we know that the age distribution at the end of the growth period minus the harvested animals will give us the age distribution at the beginning of the next growth period: Lx(t) − HLx(t) = x(t) However, by assuming certain attributes, we create a variety of problems.

Thus the Leslie Matrix is only an idealistic version of how to calculate population growth and harvest. As such, we must address these issue before hand.

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Possible Problems with Assumptions

A number of the assumptions made to use the Leslie Matrix are potentially unreasonable when considering real populations.

Assuming that the fraction of females in each age class is approximately the same as that of males is only a reasonable assumption to make with mammalian populations where sex determination appears stochastic. For other populations such as certain reptile species, sex is determined by temperature, so a particularly cold growth period might render a much higher fraction of males than females.

Assuming that there is no immigration or emigration between populations is only appropriate for isolated populations. Very rarely are populations truly isolated, especially in the case of bird populations.

The Leslie Matrix is largely limited to mammal populations that live in a relatively fertile and approximately unlimited environment such that their birth rates remain constant from one growth period to the next.

The Leslie Matrix model does not account for sporadic large-scale disturbances in a population’s environment such as disease, effects of invasive species, or natural disasters. If any of these factors play a continuous role and are reflected in the life tables collected by ecologists, then the Leslie Matrix can account for them.

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Problems

1. Let a certain animal population be divided into three 1-year age classes and have as its Leslie Matrix   0 4 3 L =  21 0 0 0 14 0 (a) Find the yield and the age distribution vector after each harvest if the same fraction of each of the three age classes is harvested every year. (b) Find the yield and the age distribution vector after each harvest if only the youngest age class is harvested every year. Also find the fraction of the youngest age class that is harvested. (a) First let’s find the fraction of of the number of females harvested from each group. We can do this by finding the first eigenvalue of the Leslie growth matrix and then the fraction of individuals harvested is equal to 1 − ( λ11 ). The first eigenvalue of the Leslie growth matrix is 32 (found using Maple). This implies that the harvesting fraction is 13 .

The yield is equal to the number of females harvested from each age group.

This corresponds to the product HLx(t):    1 0 4 3 x1 0 0 3 HLx(t) =  0 13 0   12 0 0 x2  x3 0 14 0 0 0 13 This simplifies to  1   4  4( 3 )x2 + 3( 13 )x3 ( 3 )x2 + x3 1 1 1   =⇒   2 ( 3 )x1 6 x1 1 1 1 ( )x x 2 4 3 12 2

This vector represents the yield.

The age distribution of the population after the harvest is the original age distribution matrix after the growth period Lx(t) minus the animals harvested HLx(t). This gives     1  8  0 4 3 x1 4( 3 )x2 + 3( 13 )x3 3 x2 + 2x3 1 1  1 0 0 x2  −   =  1 x1  2 2 ( 3 )x1 3 1 1 1 1 0 4 0 x3 4 ( 3 )x2 6 x2 This final matrix is the age distribution matrix after harvest.

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(b)To find the yield  f 0 0 0 0 0

we do a similar procedure as in (a). We find HLx:      0 4 3 x1 0 4f x2 + 3f x3  0  12 0 0 x2  =  0 1 0 0 0 4 0 x3

This is the yield. To find the age distribution vector we have to find Lx-HLx:       (4x2 + 3x3 )(1 − f ) 4x2 + 3f x3 4f x2 + 3f x3 1 1  −  = 0 2 x1 2 x1 1 1 0 x x 4 2 4 2 Using this equation we can create a system of equations and solve for f : 1 1 x1 = x2 x2 = x3 (4x2 + 3x3 )(1 − f ) = x1 2 4 Substituting and eliminating variables gives 1 1 3 11 4 x1 + 3 x1 (1 − f ) = x1 =⇒ 2 + (1 − f ) = 1 =⇒ f = 2 8 8 19 2. For the optimal sustainable harvesting policy described by Equation 13 (see project textbook by Anton), find the vector x1 that specifies the age distribution of the population after each harvest. Also calculate the vector x1 and verify that the optimal sustainable yield is 19.9 percent.

We calculate Lx(t)-HLx(t)=x(t). After some simplification we get 

 0.488(0.045x2 + 0.391x3 + 0.472x4 + 0.489x5 + 0.546x6 + 0.543x7 + 0.502x8 + 0.468x9 + 0.459x10 + 0.433x11 + 0.421x12      0.845x1     0.975x 2     0.965x 3     0.950x 4     0.926x 5     0.895x 6     0.850x 7     0     0.691x 9     0.561x10 0.370x11 To confirm the percent yield we can divide the number harvested individuals by the number of total individuals across all age classes. We know the percentage for the optimal yield should come out to be 19.9 yet, an error must have occurred with our derivation of Lx(t) − HLx(t) in solving for x. Ultimately we achieved the trivial solution for vectors x1 through x12 .

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3. Use equation 10 to show that if only the first age class of an animal population is harvested,   R−1  0      Lx(t)1 − x(t)1 =  0   ..   .  0 Where R is the net reproduction rate of the population.

Based on definitions given in the Anton document we have the following:      1 1 a1 a2 a3 · · · an      b1 b2 b3 b1 b1 ··· 0            b1 b2 b3 b1 b2 b1 b2 ··· 0  Lx(t)1 −x(t)1 =   −      .. .. ..     . . . b1 b2 · · · bn−1 b1 b2 · · · bn−1 0 0 · · · bn−1 0   (a1 + a2 b1 + a3 b1 b2 + · · · + an b1 b2 · · · bn−1 ) − 1   0     0 =⇒     ..   .

0 And by definition R = a1 + a2 b1 + a3 b1 b2 + · · · + an b1 b2 · · · bn−1 . So   R−1  0      Lx(t)1 − x(t)1 =  0   ..   .  0 4. If only the Ith class of an animal population is to be periodically harvest (I = 1, 2, ..., n) find the corresponding harvesting fraction hI .

For this problem we seek to have an animal population where the net reproduction rate is 1. We have a general formula for the net reproductive rate which is: (1−h1 )[a1 +a2 b1 (1−h2 )+a3 b1 b2 (1−h2 )(1−h3 )+· · ·+an b1 b2 · · · bn−1 (1−h2 )(1−h3 ) · · · (1−hn )] = 1 If we let h1 = h and all other h’s become 0 then the net reproductive rate formula simplifies to h = 1 − 1/R. If we let h2 = h, then the net reproductive rate becomes: a2 b1 (1 − h) + a3 b1 b2 (1 − h) + · · · + an b1 b2 · · · bn−1 (1 − h) = 1 1 Thus we find that the formula for h2 = h is h = 1 − a2 b1 +a3 b1 b2 +···+a . n b1 b2 ···bn−1

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However, finding a formula for each hI becomes tedious and repetitious. As such, we may generalize the formula for each hI = h to be: hI = 1 −

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1 aI b1 b2 · · · bI−1 + · · · + an b1 b2 · · · bn − 1

Interesting Examples

In “Population Consequences of Fipronil and Degradates to Copepods at Field Concentrations: An Integration of Life Cycle Testing with Leslie Matrix Population Modeling”, ecologists used Leslie Matrices in the study of Copepods to determine the effects of toxins in their environment. Ecologist collected demographic data on the larvae of the copepods, including survival, development rates, sex ratio change, fertility, fecundity, and hatching success for 32 days through mating and production of three groups in seawater spiked with toxins (Chandler). The demographic data was used to construct a Leslie Matrix to predict relative rates of population increase (using the largest eigenvalue of the matrix) and changes in net population growth with time and toxicant concentration. The results of the analysis indicated that “reproductively linked impacts of fipronil and its degradation products (the toxins) at the population maintenance levels suggest risks to sediment-dwelling crustaceans at concentrations well below noneffects for most aquatic test species based on risk assessment data from primarily acute and sub-life cycle toxicity tests”(Chandler). This means that copepods are more sensitive to toxins than other aquatic life and the declination in copepod populations can later have an impact on other “test species”(Chandler).

In this example Jensen uses matrices to model the growth of a deer population when different groups are harvested. This example is different from the example we studied with the sheep because this example with the deer population classifies groups by sex rather than age(Jensen). Jensen studies the harvesting of specifically the male population, the female population, and harvesting a mix of both males and females. The harvesting of each group provides a unique result for the growth of the total population. For instance, should only the males of the population be harvested, the female population grew and the yield increased asymptotically. Conversely, should only the females of the population be harvested, the harvest initially increased, yet dropped later on, producing a harvest that was smaller than when both sexes were harvested(Jensen).

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Conclusion

In this project we have investigated the use of the Leslie Matrix Model and how to to use it to study populations. We discussed the mechanics of the Leslie Matrix, and how the Leslie Matrix can be used to predict the age distribution of a population after a growth period. We described how many individuals, Leslie, Bernardelli and Lewis, worked toward developing the tool independently in the

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early 1940s. We discussed how to create a Leslie Matrix from demographic data about a population, and what relevant Linear Algebra topics we use when we talk about Leslie Matrices, including dot products of matrices and vectors and eigenvalues. We discussed assumptions about the Leslie Matrix and how it best applies to high density mammalian populations. Finally, we considered a number of problems from the Anton textbook and derived some properties about the Leslie Matrix given specific harvesting practices.

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Citations • Anton, Howard, and Chris Rorres. Elemetary Linear Algebra. 10th ed.

S.l.: J. Wiley, 2005. Print.

• Beattie, Joshua S. Methodology of Science: Biology. 2012. Course Material. Stanford University Online High School, Stanford.

• Chandler, G. Thomas, Tawnya L. Cary, Adriana C. Bejarano, Jack Pender, and John L. Ferry. “Population Consequences of Fipronil and Degradates to Copepods at Field Concentrations: An Integration of Life Cycle Testing with Leslie Matrix Population Modeling.” Population Consequences of Fipronil and Degradates to Copepods at Field Concentrations: An Integration of Life Cycle Testing with Leslie Matrix Population Modeling - Environmental Science Technology (ACS Publications). University of South Carolina, 31 Aug. 2004. Web. 21 Nov. 2016.

• Jensen, A. L. “Sex and Age Structured Matrix Model Applied to Harvesting a White Tailed Deer Population.” Sex and Age Structured Matrix Model Applied to Harvesting a White Tailed Deer Population. Elsevier B.V, 20 Apr. 2000. Web. 21 Nov. 2016.

• Jones, James Holland. Leslie Matrix I Formal Demography Stanford Spring Workshop Web.Stanford.edu. Department of Anthropology, Stanford University, 2 May 2008. Web. 16 Nov. 2016.

• Kiser, Klaus. Leslie Matrices. Reston, VA: National Council of Teachers of Mathematics, 1988. Math.uh.edu. University of Houston. Web. 20 Nov. 2016.

• Leslie, Patrick H. “On the Use of Matrices in Certain Population Mathematics.” Biometrika 33.3 (1945): 183-212. Web. 17 Nov. 2016.

• Michigan State University. Leslie Matrix. East Lansing: Michigan State University, 2013. PPT. • “Modeling Population Growth.” Modeling Population Growth. Trans.

William F. Farr. Worcester Polytechnic Institute, 24 Sept. 1998. Web. 21 Nov. 2016.

• Quinn, Terrance J., and R. B. Deriso. Quantitative Fish Dynamics. New York: Oxford UP, 1999. Print.

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