Rise of the Feral Pigeons: Introduction to Leslie Matrices

We've seen in applied mathematics, there are several models for population growth of a species. In my own posts, I have briefly covered the natural (or exponential), logistic growth and predator-prey models.

So far, all of these models treat the population(s) in question as a homogeneous species. They don't take into account most species have 2 sexes (male and female), different age classes within the population and how reproductive rates may differ between them, nor any other factors that may affect the structure of a particular population.

It is intuitive to think that birth and death rates varying according to age.

For instance, we know that in our own human population, that women who are aged under 12 are much less likely to give birth to say women between the ages of 24 and 36. Women aged 37 to 50 are also less likely to give birth than the aforementioned group.

We also know that generally, you are less likely to die if you are younger. It is a fact of life that as a population gets older, the rates of death increase too. (Of course, there are some animals that oppose this trend, e.g. turtles as hatchlings have a much higher death rate due to predators, but those who survive go on to live very long lives).

So is there a mathematical model that can take some of these factors into account? Yes, there is!

Feral Pigeons running rampant in the City

Let's jump right into an example...

Figure 1. Feral Pidgeons in Trafalgar Square
Image Source: Wikimedia Commons

Over some time, data was collected by zoologists on feral pigeons living and breeding in a city. A snapshot of current age-structured population of the feral pigeons is detailed in Table 1.

Age (years)	0 - 1	1 - 2	2 - 3	3 - 4
Population	2500	1000	300	150
Birth rate	0	3.5	1.5	0.5
Death rate	0.6	0.5	0.7	1

Table 1. Feral Pidgeon Population Data
Can we use this data to predict the population next year? Yes, we can!

First of all, we have the birth rates for each age class, so we can use this to predict how many new born feral pigeons will come into existance in the 0 - 1 age class at the end of the cycle (i.e. 1 year). We simply use the formula...

ΣNew borns (Age: 0 - 1) = Σ(Birth rate × Population in Age Class)

Table 2 below is the calculation for new pigeons that come into existence at the end of the cycle.

Age: 0 - 1

0x2500

+

3.5x1000

+

1.5x300

+

0.5x150

= 4025

Table 2. Number of new-born pidgeons
The population is also increased by the number that survive from one year to the next. We've been given their death rates (i.e. the portion that die every year for each age class). Therefore the portion that survive and advance to the next age class is given by:

Survival rate = 1 - Death rate

The survival rates according to age class are shown in Table 3.

Age	0 - 1	1 - 2	2 - 3	3 - 4
Survival rate	0.4	0.5	0.3	0

Table 3. Survival rate according to age class
We can calculate the surviving population and add it to the new borns from Table 2. This is shown in Table 4.

Age: 0 - 1	0x2500	+	3.5x1000	+	1.5x300	+	0.5x150	= 4025
Age: 1 - 2	0.4x2500							= 1000
Age: 2 - 3			0.5x1000					= 500
Age: 3 - 4					0.3x300			= 90
							Total	= 5615

Table 4. Calculating total population after 1 cycle
Note: in table 4, it is not necessary to calculate how many pigeons are aged 4 - 5 because we know from Table 1, they all die!

Thus at the same time next year (end of 1 cycle), we can expect the total number of feral pigeons in the city to be 5615, a whopping 42% increase. This is clearly going to be a problem that we'll have to deal with, and we'll get to that later.

But firstly, notice how the operations in Table 4 can be carried out as a Matrix-Vector multiplication.

Let X₀ denote the initial population vector...

Let L be the "birth and survival" matrix.

L is called the Leslie matrix, invented by and named after Patrick H. Leslie. The Leslie model is commonly used in ecology, where age-structured changes in a species population are important. It conveniently generates population predictions in age classes.

Ok, let's verify that we get the same numbers after year 1 when we carry out the matrix-vector operation...

Great. We have matching numbers from Table 4 and population vector X₁. So the Leslie matrix gives us a much more convenient way to calculate our predictions.

Assuming the conditions for the feral pigeons survival remain constant year on year, we can confidently make projections about their population well into the future. Here's the calculation for population X₂, 2 years after the initial snapshot...

The total population from X₂ is 4295+1610+500+150 = 6555, which is an increase of 16.7% from X₁.

After 3 years...

The total population from X₃ is 9133, a 39.3% increase from X₂.

After 4 years...

The total population from X₄ is 10980, a 20.2% increase from X₃.

I think you're getting the picture that we can iterate this operation indefinitely. A 22 year prediction for city pigeon population is shown in Figure 2 below. The growth is reminiscent of the exponential model.

Figure 2. Growth in Feral Pidgeon population over 22 years

Now, of more interest is the percentage change in population year by year shown in Figure 3.

Figure 3. Percentage increase in Pidgeon population year by year

The population is increasing, and thus the percentage change is always positive. But the growth rate fluctuates wildly between the 2 blue dashed regression lines in the early years before it converges to constant growth of about 28.5% per year. Knowing this will be important in my next post on strategies for population control.

Finally, since age-structure is one of the hallmarks for the Leslie model, let's have a look at how the population is spread across the age-classes as time passes on...

Figure 3. Distribution of Pidgeon population across age classes

Thus we can see that eventually, the population distribution settles to about...

• 68.4% aged 0 - 1
• 21.4% aged 1 - 2
• 8.2% aged 2 - 3
• 2% aged 3 - 4

In the next post, we'll look at mathematical strategies for population control.

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Credits:

All equations in this tutorial were created with QuickLatex

All graphs were created with www.desmos.com/calculator

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