Vishvas's notes

Notation

Let us revise and adopt the following notations: b is the number of offspring per generation per individual of the parent generation, d is probability of death of an individual per generation and hence R = 1 + b − d. With this notation,the model we described in the class which accounts for reduced resources with increased population has the form $R = R_{0} (1 - N_{t} / K)$ and hence, $N_{t + 1} = R_{0} N_{t} (1 - N_{t} / K)$ .

1

You observe that a population of dragonflies show oscillatory patterns, i.e. population density is changing roughly periodically with time. Propose two hypotheses- with a brief reasoning for each of them - for the observed patterns.

Response

Changing seasons could be affecting food availability, leading to alternating increases and decreases.
An inter-species competition with either species gaining the upper hand for a while before the other adapts.
A predator prey cycle. Like the hare-lynx model in book chapter “Modelling Philosophy” by Hanna Kokko.

2

What is the biological interpretation of $R_{0}$ and $K$ in the population equation $N_{t + 1} = R_{0} N_{t} (1 - \frac{N_{t}}{k})$ ?

Response

$R_{0}$ is max(R).

k measures the living room or resources available - like Oxygen for Human apes. If k is $\infty$ , $N_{t}$ grows at max rate. If k is $N_{t}$ , everyone dies.

3

Consider $R = e^{r (1 - \frac{N_{t}}{k})}$ , i.e. an exponential decay. The population model now reads: $N_{t + 1} = N_{t} e^{r (1 - \frac{N_{t}}{k})}$ .

3a

What is the equivalent of $R_{0}$ , and what is a possible biological interpretation of K in this model?

Response

$e^{r}$ is the equivalent of $R_{0}$ , the max rate of growth. If k is $\infty$ , $N_{t}$ grows at max rate. If k is $N_{t}$ , numbers stay constant. So, $k$ is max numbers that the environment can support.

3b

This model too has two parameters, r and K. Eliminate one of the variables without loosing any information and write the resulting simplified model.(10 Marks).

Response

Dividing both sides by k, we get $\frac{N_{t + 1}}{k} = \frac{N_{t}}{k} e^{r (1 - \frac{N_{t}}{k})}$ . Scaling to get $x_{t} = \frac{N_{t}}{k}$ , we get $x_{t + 1} = x_{t} e^{r (1 - x_{t})}$ .

4

Bacteria has a generation time of 20 minutes at the end of which it divides to give rise to daughters when sufficient resources are around. A student puts a bacterium in a container having sufficient resources for reproduction and then she ensures that per-capita resource availability never goes down. If she indeed maintains that condition, what will be the number of bacteria and the approximate weight of the beaker at the end of 36 hours? For this question: (a) estimate numbers manually without using calculators/computers (b) list any assumptions you make clearly.

Response

Let w be the weight of a bacterium. In 36 hours, there will be 108 generations. So, there will be $2^{108}$ bacteriae, whose weight will be $2^{108} w$ , or over $10^{30} w$ . This means that if w = 1 picogram, the weight will be over $10^{1} 5$ kgs. Very heavy.

Assumptions:

Each bacterium splits into 2 when reproducing.
The beaker is at least as big and strong as a three storey building, and that the student has access to helicopter and crane to keep supplying food to the bacteria - though I wonder if even that would suffice.

5

Required readings: (1) Read the uploaded book chapter “Modelling Philosophy” by Hanna Kokko. (2) Strategy of modelling in Ecology by Levins (first two and a half pages).

5(a)

Write three key criteria you would use to evaluate a mathematical model of a biological system.

Response

Does it answer some clear causal questions? (of the type: “Under conditions A, X leads to Y.”) Or does it intend to make quantitative predictions?
Is it sufficiently abstract (and therefore intelligible), and not loaded down with unnecessary details?
Is it mathematically elegant (in a way that could contribute to building mathematical theories for their own sake)?

5(b)

Levins talks about the importance of ’robust’ results which are independent of the detailed assumptions. What do you think are cons of having such robust results when one tries to interpret real data?

Response

Such robust results might blind one to other causes for the same observation. The expectation to see a “robust causal relationship” might cause one to neglect a new factor in the environment. For example, seeing a decline in a population might cause one to look for increases in predator population while missing new factors such as road traffic or pesticides.

Also, such robust results, being mostly qualitative, sacrifice precision - meaning they cannot make good quantitative predictions.