Unit 4: Population Growth Flashcards
What is the definition of a population? What is population size? What does it depend on?
Population: All of the individuals of a given species that live and reproduce in a particular place.
This particular place can be very vague, but in general we want it to be a specific area where a group lives.
Population size is the number of individuals alive at a particular time in a particular place and is influenced by births, deaths, immigration and emigration. So population changes over time and hence this is just a snapshot.
We don’t look at migration here.
If ignoring migration, what does the change in the total number of individuals in a population depend on?
The change in the number of individuals depends on births and deaths, so B - D is approximately equal to the change in population size over a given amount of time.
What is per capita birth and death rate? How do we track this? What is per capita growth rate? How can this equation be manipulated?
Per capita birth rate is the number of births per person in a population per unit time (is each individual replicating itself and more, or a fraction of itself?) So how many people are added to the population per population size?
For example, if 45 people are added per year, and there are currently 450 people, then the per capita growth rate is 0.1, meaning each individual is producing 0.1 individuals on average. This is calculated by dividing the number of births in a year by the populations adult population. If it is zero, that means that all offspring died before they could reproduce. If it is greater then 1, then each individual is replicating themselves and producing more, leading to growth. So on average, this gives how many births are occurring per individual.
Per capita death rate is the number of deaths scaled to the size of that population over a given amount of time. So it is essentially the number of people dying per person alive over a given amount of time. If this is greater then 1, then for every person alive more then them are dying resulting in a decaying growth rate.
Per capita growth rate is then the change in population size over time, divided by population size currently (original population size before this new generation is added). So how many people are added to the population over time (net because excluding deaths) per individual already present. This is calculated by the number of births minus the number of deaths, all divided by population size. Or the per capita birth rate minus the per capita death rate, which just turns into this formula due to common denominator. because this indicates how many are being added per individual, we can determine how the population is growing. If r>0, the population is growing because the birth rate is larger than the death rate. If r<0, the birth rate is smaller then the death rate and the population is shrinking. This is similar to Ro from the life history tables, except for that one if it was greater then 1 it would be growing and less then one it would be shrinking.
Then to manipulate this, per capita growth rate is also equal to growth rate times population size. (R times N) This is because the change in population size over time (dN/dt) is equal to births minus deaths. So then since the per capita growth rate is B-D/N, this is also (dN/dt)/N. Multiply both sides by N, and rN is equal to dN/dt. This makes sense because the number of individuals present times the growth rate will give the number of individuals that were added to the population, which is the rate of change of the population over time.
How do you calculate change in population growth? What does r stand for? What about N o?
To calculate change in population growth, you do dN/dt = rN o. Because this is the number of individuals being added over time, as you are multiplying the growth rate fraction by the population size currently (which is No when doing this population growth). if positive, then the population is growing because you are adding a given number of individuals over time. If negative, you are taking away a given number of individuals over time.
Since r = b - d = (B - D)/N, then dN/dt = (B-D)/N x N = B - D which makes sense, as this is how many are being added.
What is the exponential model of population growth? What does the equation look like for this model?
The exponential model is essentially how the population would grow if it had no limiting factors — in a perfect world. Essentially, the population growth rate is rmax, and it never changes from this value. Therefore, the population just keeps exponentially increasing as the rmax is applied to the N of each generation, resulting in more and more individuals being added each time.
Therefore equation for this graph is y=ab^x, where b is the growth factor, a is the original population number, and x is the number of generations that have occurred. So you multiply a by that growth factor however many times and this accounts for compounding!
We have learned how to calculate rNo or the growth rate over a certain amount of time. But what about modelling the population size over time? How would this be shown?
To do this, we would use the equation Nt= No(1 + rmax)^t. And here rmax is constant!
Then Nt = population size at a certain point in time
No = the original population number
Rmax = the maximum rate of growth with no limiting factors
T= the amount of generations that have passed.
Intuitively this makes sense, because we are multiplying No by itself (that’s where the 1 comes in) plus rmax, which accounts for how it is growing. Then t is the number of generations this growth was applied to, and since it is exponential this accounts for the fact that No is different each time! Because you multiply No by the growth factor, and then you multiply that new value by the growth factor again, and then that new value by the growth factor again. This is what exponential patterns do, and this allows for the changing population size over time to be accounted for.
Determine how many individuals are present in generation two of this population:
B = 80
D = 60
No = 100
T=1
T=2
So we could calculate 1 and then 2, but since rmax is constant we really only have to do 2.
rmax = (B - D)/100 = 0.2 (so this is the proportion of the current population that is added each generation).
Then N2 =No(1+rmax)^t
N2 =100(1+0.2)^2
If we square it, essentially we multiply 1.02 times 100 to get the first generation, then multiply this by 1.02 again to get the second generation. This is why we can do it all at once.
N2 =100(1+0.2)^2 =144
So over the first two generations, 44 individuals were added.
Is r constant for the exponential growth model? If so, why does it keep increasing by different amounts over time? Will it ever reach a maximum?
Yes r is constant, because it is always growing at its maximum capacity. HOWEVER, the population size is not constant, and r is always multiplied by the current population size. therefore, because there are more individuals produced each generation, the number of individuals added each time keeps increasing, but the FRACTION of individuals remains the same. It’s just that the number that you are multiplying that fraction by keeps increasing, resulting in exponential growth. This is called a compounding relationship.
Exponential growth will never reach a maximum if nothing limits it, however rmax will be at its maximum.
Why is the exponential model not a realistic model in nature? What can it be used for?
This is because resources and predators usually limit the growth of organisms, and therefore rmax cannot always be present. There is a carrying capacity in nature, and if organisms just keep being added at an exponential rate, it will not be sustainable.
However, this model can be used for short term rapidly growing regions who are not near carrying capacity and hence represent this model.
It can also be used as a comparison to actual population growth to see how the lack of resources in the area may be effecting that growth.
What are the 6 main factors that limit population growth? What do these factors contribute to?
-temperature
-precipitation
-disease
-food availability
-Sunlight
-population of predators
These factors contribute to the carrying capacity of the ecosystem, which is essentially the maximum population size that can be supported based on the resources available and all these other factors. This is why the exponential growth model is not realistic over long term (though it may be used after a wipe out of a population).
In reality, how does r change as a population approaches k? What graph reflects this relationship?
In reality, as the population approaches k, r will decrease from rmax, because there are less resources available for all these organisms to compete and survive. So growth rates decrease and death rates increase, resulting in a decreased rmax.
So the closer a population gets to the carrying capacity, the lower the r value becomes, until it is constant at carrying capacity.
Overall: As N > K, r will decrease due to more competition, less births and more deaths, and therefore growth rate decreases.
What does a graph of per capita growth rate VS population size look like? When does K=N on this graph and why? What does the negative population growth rate mean below this point?
As per capita growth rate increases, population size decreases, and vice versa. This is because as the growth rate increases, more individuals are added to the population and this cannot be supported by the resources present.
So if you look at this from one way: As population size approaches K, the per capita growth rate must decrease in order to flatten out the graph and match the resources present (competition and lack of resources as population size increases will decrease births and increase deaths and lead to a decreased r).
Or, as per capita growth rate increases, the population size much decrease to keep the population at carrying capacity. Because if you are multiplying an increasing growth rate by a decreasing population size, your population should not exponentially grow, it should slow down instead.
Where this graph intercepts the x-axis is where N=K, because this is the equilibrium point that most populations will be ending up at. So once you surpass this point, growth rate becomes negative because there is not enough resources to sustain the population and therefore deaths must overtake births to bring that size back to equilibrium. When it is below the equilibrium value, the population growth rate will be positive because you want to add as many individuals as possible to prosper if the environment allows for that. But as you approach equilibrium, r will decrease because less individuals can be added, until it reaches 0 at the K (no net organisms will be added).
What is the logistic model of population growth? What does it account for and what does the graph of population over time look like for this one? What does the POI represent?
The logistic model of population growth is a model which accounts for a changing r value as the population approaches carrying capacity, and so it does not assume ideal conditions, it assumes there are many limiting factors to that population’s growth.
Essentially, as the population size grows, death rates increase and birth rates decrease due to less resources and food and more competition. This also makes them weaker so predators can more easily hunt them.
Once the population size reaches carrying capacity, r is pretty much zero because the net growth rate is 0 (a constant amount of individuals are born and die over time).
This graph is a sigmodial curve (s-shaped curve). Essentially, it looks like the exponential growth model at first, because there is not much limitations due to a small population size and hence r stays very close to r max resulting in rapid growth. But because the No’s are very small, this growth is not as exponential as it is at the far end of the exponential growth model graph. SO when the growth rate is at its highest, the actual increase in individuals is at its lowest at first because you are multiplying the growth rate by such a small population size.
Then once it increases exponentially for a long enough time, N reaches half of the carrying capacity, and this is where the population size is increasing by the largest value, since it maximizes the r value AND the population size. So the r value is multiplying by a large enough N value that growth can occur very rapidly. But this is the largest that growth gets, because past this point N is too large and so r must drastically decrease, resulting in a concave down section where r decreases until it reaches 0. This 0 point is where the carrying capacity is, and so on average no net amount of individuals are added or taken away since the population size is maximized for the given resources.
So essentially the POI is just the maximum efficiency of growth and maximum rate of growth, where r and N can be maximized together to add the largest number of individuals to the population at that given time.
How does r change as the N approaches K, and what is the formula to calculate r at each generation?
R decreases linearly as N increases and approaches k, and this is represented by:
Rt= rmax ((K-Nt)/K)
So this is r max times some factor that specifies how close it is to k.
Rt will be one less then the generation you are solving for, because its the growth rate between the one you are currently at and the one you are trying to get to. So if you want to find the population size one year after the original measurement, No = initial population size. N1 = new one you are solving for. And then ro is what you want to find to know how the population changes from No to N1.
So you would do ro = rmax ((K-No)/k)
This makes sense, because the growth rate from one gen to the next depends on the current gen size, so you want No and r to be the same subscript.
What does the formula look like for calculating population size, and how does this process differ from the exponential growth model?
The exponential growth model is a constant calculation of Nt= No(1+rmax), where No is always the original and r is always rmax.
But with logistic growth, r changes over time, and so for each generation we must recalculate r to get this equation.
Therefore, it looks like this: Nt+1 = Nt(1+rt). So the next generation is equal to the current generation size multiplied by the growth rate for that current generation, which is also dependent on its distance from k.
So for calculating N at any given time for this model, you first must calculate what r would be going from that generation to the next, using rt=rmax ((K-Nt)/K). Then you use the formula: Nt+1 = Nt(1+rt) to get the population size for the next generation.
In this case, you cannot just use the Nt=No(1+rmax)^t, and put in the t value to solve for any generation in one step. This is because r changes for each generation! So to solve for the 3rd generation of logistic growth, you must find the r value between original and 1, 1 and 2, and 2 and 3, and then do the Nt+1 = Nt(1+rt) for each generation! No exponents can be used for simplification!
