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<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>Welcome to Population Ecology!</title>
<meta charset="utf-8" />
<meta name="date" content="2021-04-08" />
<script src="libs/header-attrs/header-attrs.js"></script>
<link href="libs/remark-css/default.css" rel="stylesheet" />
<link href="libs/remark-css/default-fonts.css" rel="stylesheet" />
</head>
<body>
<textarea id="source">
class: center, middle, inverse, title-slide
# Welcome to Population Ecology!
## ⚔<br/>
###
### 04/08/2021
---
# 3 part lecture (with breaks)
.pull-left[
- **Part 1** Exponential growth
- **Part 2** Logistic growth - _Density dependent_
- **Part 3** Logistic growth with stochasticity
]
.pull-right[
]
---
# Part 1: Why be interested in population growth?
--
- Project future populations
- Human population expected to be 9.8 billion by 2050
--
- Conservation of species
--
- Sustainable use of resources
--
- Many more
---
class: inverse, middle, center
#Part 1: Let's start with exponential population growth
Density independent growth
`$$\frac{dN}{dt}=rN$$`
---
#Exponential growth
What is required for a population to grow?
--
How many births and how many deaths?
`$$N_{t+1} = N_t + B - D + I - E$$`
- `\(B\)` = _Births_
- `\(D\)` = _Deaths_
- `\(I\)` = _Immigration_
- `\(E\)` = _Emigration_
--
If we assume that immigration and emigration are equal, then the change in population size is:
`$$\Delta N = B - D$$`
---
#Exponential growth
`$$\Delta N = B - D$$`
.pull-left[
- More births than deaths the population grows
.center[
<img src="pop_growth_files/happy_face.png" style="width: 50%" />
]
]
.pull-right[
- More deaths than births the population declines
{{content}}
]
--
{{content}}
---
#Exponential growth
_Change in population_ ( `\(dN\)` ) _over a very small interval of time_ ( `\(dt\)` ) can be described as:
`$$\frac{dN}{dt}=B-D$$`
--
- Births and deaths are also described in rates:
.pull-left[
`$$B = bN$$`
`\(b\)` = instantaneous birth rate
[births / (individual * time)]
]
.pull-right[
`$$D = dN$$`
`\(d\)` = instantaneous death rate
[deaths / (individual * time)]
]
---
#Exponential growth
_SO_ change in population over time can be described as
`$$\frac{dN}{dt}=(b-d)N$$`
--
`$$\frac{dN}{dt}=(0.55-0.50)N$$`
--
`$$\frac{dN}{dt}=(0.55-0.50)*100$$`
--
`$$\frac{dN}{dt}=0.05*100$$`
--
`$$\frac{dN}{dt}=5$$`
---
#Exponential growth
If we let `\(b-d\)` become the constant `\(r\)`, the __intrinsic rate of increase__, we have the continuous exponential growth equation:
`$$\frac{dN}{dt}=rN$$`
--
.center[
<img src="pop_growth_files/ta_da.png" style="width: 60%" />
]
---
#Exponential growth
Change in population is equal to the intrinsic rate of increase ( `\(r\)` ) multiplied by the population size ( `\(N\)` )
.pull-left[
`$$\frac{dN}{dt}=rN$$`
]
.pull-right[
`\(N\)` = _population size_
`\(r\)` = _intrinsic rate or increase_
]
`\(r\)` defines how fast a population is growing or declining
- `\(r=0\)` no growth
- `\(r>0\)` positive growth
- `\(r<0\)` negative growth
The differential equation tells us growth __rate__ not population size
---
#Exponential growth
The abundance of an exponentially growing population at a given time `\(t\)` can be worked out by:
`$$N_t = N_0e^{rt}$$`
Look familiar from assignment 1?
--
The discrete version of the exponential equation tells us the population per time-step:
`$$N_{t+1} = N_t + r_dN_t$$`
`$$N_{t+1} = 100 + 0.05 * 100$$`
`$$N_{t+1} = 105$$`
`\(N_t\)` = Population size at time t
`\(r_d\)` = discrete growth factor
---
#Exponential growth
.pull-left[
Theoretical populations growing and declining as a result of different values of `\(r\)`
- `\(r=0\)` no growth
- `\(r>0\)` positive growth
- `\(r<0\)` negative growth
Because growth rate is exponential, by taking the natural logarithm of the population size the graphed lines become straight
]
.pull-right[
<!-- -->
]
---
# Exponential growth
.pull-left[Exponential growth]
.pull-right[Logarithm of exponential growth]
---
#Growth rates
Common name | `\(r\)` [individuals/(individual*day)] | Doubling time
------------|-------------------------------------|---------
Virus | 300.0 | 3.3 minutes
Bacterium | 58.7 | 17 minutes
Protozoan | 1.59 | 10.5 hours
Hydra | 0.34 | 2 days
Flour beetle| 0.101 | 6.9 days
Brown rat | 0.0148 | 46.8 days
Domestic cow| 0.001 | 1.9 years
Mangrove | 0.00055 | 3.5 years
Southern beech | 0.000075 | 25.3 years
---
#Growth rates
Population increases exponentially but __Growth rate__ over __population size__ increases proportionally with the population
---
# Exponential growth
- Populations growing exponentially have a doubling time
- The doubling time depends on the growth rate `\(r\)` and is _not_ every year
- (ง'̀-'́)ง Surely no species can grow forever exponentially?!?!?!
- _Correct!_ Welcome to part 2, __density dependence__
---
class: inverse, middle, center
#Take 5 minutes to discuss the assumptions of the exponential growth model
( ͡° ͜ʖ ͡°)
---
class: inverse, middle, center
#Part 2: Logistic population growth
Density dependent growth
`$$\frac{dN}{dt}=rN\left(1- \frac{N}{K} \right)$$`
---
#Logistic growth
Now we will look at populations which do not grow forever but reach a __carrying capacity__ ( `\(K\)` )
`\(K\)` represents the maximum population size that can be supported considering limiting factors such as food, shelter and space
`$$\frac{dN}{dt}=rN\left(1- \frac{N}{K} \right)$$`
`\(N\)` = Population size
`\(r\)` = Intrinsic rate of increase
`\(K\)` = Carrying capacity
---
#Logistic growth
Consider the term `\(\left(1- \frac{N}{K} \right)\)` as a penalty on the growth of the population depending on the number of individuals in the community
- Very crowded communities have a high penalty compared to ones with plenty of space and resources
`$$\frac{dN}{dt}=rN\left(1- \frac{N}{K} \right)$$`
--
- At the point the population reaches the carrying capacity: `\(N\)` will = `\(K\)`, and the fraction `\(\frac{N}{K}\)` will = 1
- The term `\(\left(1- \frac{N}{K} \right)\)` will collapse to 0 and the change in population will = 0. The equation will be multiplied by 0 and equal 0
- So the population will remain at size `\(K\)`
---
#Logistic growth
If the population is very small, `\(N\)` is small relative to `\(K\)`, then the penalty is small
`$$\frac{dN}{dt}=rN\left(1- \frac{N}{K} \right)$$`
--
However, as we learned from exponential growth, a population grows in proportion to its size.
- A population of 1000 seabirds will produce more eggs than a population of 100.
- In the logistic growth equation the proportion added to the population decreases as the population grows, reaching 0 when `\(N = K\)`
---
#Logistic growth
`$$\frac{dN}{dt}=rN\left(1- \frac{N}{K} \right)$$`
- If `\(K = 100\)` and `\(N = 7\)` then the unused space is `\(1-(7/100) = 0.93\)` and the population is growing at 93% of the growth rate of an exponentially growing population
- If `\(K = 100\)` and `\(N = 98\)` then the unused proportion of capacity is `\(1-(98/100) = 0.02\)` and growth is at 2% of the growth rate of an exponentially growing population
--
The point at which a population is the largest relative to the penalty for its size is at `\(K/2\)`
- What this means is the growth rate of a population is fastest at half its carrying capacity
- As the population exceeds `\(K/2\)` the penalty increases but below `\(K/2\)` the population is small and the proportion added to the population is also small
---
#Logistic growth
_Thanos should have studied population ecology_
.pull-left[
- If a species is at capacity, its growth rate will increase to maximum if you cut it in half.
- Not all species are equal so cutting all in half will not have an equal effect...
]
.pull-right[
<img src="pop_growth_files/thanos.jpg" style="width: 100%" />
]
---
#Logistic growth
.pull-left[
_Paramecium_ growing to capacity.
<img src="pop_growth_files/paramecium.jpg" />
]
.pull_right[
<!-- -->
]
---
#Logistic growth
.center[
<img src="pop_growth_files/yeast_seal.png" style="width: 60%"/>
]
---
#Logistic growth
Now we will look at the discrete form of the equation:
`$$N_{t+1} = N_t + r_dN_t\left(1- \frac{N_t}{K} \right)$$`
`\(N_t\)` = Population size at time t
`\(r_d\)` = discrete growth factor
`\(K\)` = Carrying capacity
--
- Instead of telling us the change in a population at an infinitely small point in time, the discrete equation tells us the size of the population at a given time.
---
#Logistic growth
The size of the population at the next time-step is equal to:
- The size of the current population plus the current population multiplied by the discrete growth rate and the density penalty.
`$$N_{t+1} = N_t + r_dN_t\left(1- \frac{N_t}{K} \right)$$`
--
`$$N_{t+1} = 100 + 0.05 * 100 \left(1- \frac{100}{200} \right)$$`
--
`$$N_{t+1} = 102.5$$`
--
Remember how in the exponential model `\(N_{t+1} = 105\)` ?
---
#Logistic growth
__Unlike__ the exponential model, in te logistic model, growth rate is dependent on population size and reaches its peak at half of the carrying capacity
---
#Logistic growth
.pull-left[
**Logistic** growth rate vs population size
]
.pull-right[
**Exponential** growth rate vs population size
]
---
class: inverse, middle, center
#Take 5 minutes to discuss the assumptions of the logistic growth model
(ᵔᴥᵔ)
---
class: inverse, middle, center
#Part 3: Introducing stochasticity
Non-deterministic growth
---
#Stochasticity
Until now, everything we have looked at has been _entirely_ deterministic but is this true of the real world?
`$$N_{t+1} = N_t + r_dN_t\left(1- \frac{N_t}{K} \right)$$`
.pull-left[
Environmental stochasticity
- Populations go through good and bad times and population growth rate is not constant
- We can represent this by adding variability to the growth rate `\(r_d\)`
]
.pull-right[
Demographic stochasticity
- By chance a population might have a run of births or a run of deaths
- Demographic stochasticity includes the probability of births and deaths in the parameter `\(r\)`
]
--
- Variability can also be included in `\(K\)`, the carrying capacity!
We will focus on _environmental stochasticity_
---
#Stochasticity
Even if __average__ growth rate is positive some stochasticity can drive extinction
---
#Recap
We have looked at:
- discrete and continuous equations for __exponential growth__
- also known as __density independent__ growth
- discrete and continuous equations for __logistic growth__
- also known as __density dependent__ growth
- __Stochastic__ logistic growth
- We have discussed model __assumptions__
---
#Key points
- __Exponentially__ growing populations grow proportional to their size indefinitely
- __Logistically__ growing populations experience a penalty that increases with population size
- Growth rate is greatest at half the carrying capacity, when the population size is at its largest relative to the density penalty
- __Stochasticity__ can have a serious effect on populations and deterministic models do not account for stochasticity
- __Assumptions are key__, _think about these for the assignment_
---
#Things to be aware of
- Differences in continuous and discrete equation
- Different notation and descriptions of `\(r\)`
- Mathematical differences
- Many different population growth equations
- We have not talked about the model `\(N_{t+1} = \lambda N_t\left(1- \frac{N_t}{K} \right)\)`.
- Concepts can be transferable but be aware results may not be the same
- This is just the beginning
- Population growth models get _much much_ more complicated to deal with more complicated species, for example, age structured population and competition
---
# Top tips for the assignment
- Attend labs and office hours
- Understand the two key graphs: population vs time and growth-rate vs population
- Use the app to understand different parameters (but do not use screenshots from the app for your assignment)
- Rubbish in rubbish out. If model outputs look crazy or the numbers don't make logical sense then double check your inputs and equations
---
class: inverse, middle, center
#Go forth and model!!!
\ (•◡•) /
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