/qa/ - Question & Answer

will have to do a 10minutes presentations on a subject i choosed i already have the script but idk if its good can someone just read it and tell me if they understand ?
(I would be very thankful if you do so because i have no one to ask, also if you ave any question on it ask me)


In the forests of the United Kingdom a century ago. In every park every wood a small red mammal with tufted ears lived. Today the red squirrel has almost disappeared from England and Wales. It has been replaced almost entirely by its North American cousin the grey squirrel introduced in 1876. The red survives in a few isolated pockets: Northumberland the Scottish Highlands and remarkably some pine forests where another animal lives. I will come back to that.
This decline is a mystery for those who do not look closely at it. The two species do not fight directly there is no predation between them and yet everywhere the grey settles the red disappears within fifteen to twenty years
My question is therefore the following <<How can mathematics model the dynamics of an animal population and predict its evolution to the point of explaining the disappearance of a species?>>
I will answer it in three points. First the Malthus model free growth when everything goes right then the Verhulst model the reality of limited resources which explains why the grey takes the advantage and finally the Lotka-Volterra model a third actor enters the scene: the pine marten whose presence reverses the situation in favour of the red.


We are in the 1890s the grey squirrel has just been introduced into a large English park. The forest is dense the acorns abundant and there are almost no predators. The population is free to grow. That is exactly what Malthus modelled in 1798. Each individual produces on average a fixed number of descendants at each generation. If we note "u(n)" the number of individuals at generation number n and r the multiplication rate of the species we have: write on the board u(n+1) = r × u(n) this is the recurrence formula which means that at a given moment n we can estimate the population at the next unit of time while its explicit expression is write on the board
u(n)= u0 × r^n
Here u0 represents the population at the start this form allows us to calculate the population for any generation simply by replacing "n" with the desired generation.
We can see that the behaviour of the population depends on the value of "r"
If r<1 the population decreases until it disappears
If r=1 the population remains stable
If r>1 the population grows without limit
In our case the grey squirrel has a reproduction rate (r) higher than that of the red therefore in an ideal environment grey populations would increase more rapidly. But this model has a fundamental flaw it assumes that "r" is constant regardless of the population density whereas in reality the resources acorns hazelnuts and space are finite. The denser the population the stronger the competition this is where Verhulst comes in.


In 1838 the Belgian mathematician Pierre-François Verhulst proposed a correction the idea is simple: every environment has a maximum carrying capacity we call it "K" the maximum capacity.
write on the board u(n+1) = r × u(n) × (1-u(n)/K)
The difference between the two models is here circle "(1-u(n)/K)" we see that the larger this part is the closer the result is to the Malthus formula and the smaller it is the smaller the population increase becomes. When we look more closely we see that the variable is the ratio u(n)/K and the larger this ratio is the smaller the population increase becomes. Since this ratio increases as the population increases according to this model the sequence is destined to converge towards a defined value which translates into a population eventually stabilising at a certain number of individuals which we will call u°.

To find u° we notice that u° is a fixed point that is to say a value of a function that does not change when the function is applied to it thus u°= r × u° × (1 - u°/K) we solve and find two solutions: u° = 0 this is extinction and u° = K × (r - 1)/r a population that does not disappear converges towards the second fixed point. Now let us return to our squirrels the red and the grey do not live in separate forests they share the same territory and the same resources therefore they share the same "K". The problem for the red is that it is less competitive for resources. The grey squirrel is larger and carries a parapoxvirus the squirrelpox. The grey develops no symptoms but for the red the disease is fatal in nearly 100% of cases. The red therefore loses on two fronts at once food competition and disease pressure. Mathematically this translates into a red population reduced in the presence of the grey its equilibrium population also decreases as the greys occupy space and transmit the disease. In regions without dense conifer forests this fixed point tends towards zero the red becomes locally extinct but this model still describes only one species at a time. To model an interaction between three actors we must go further this is where the pine marten appears a small carnivorous animal.

There are forests in the United Kingdom where the red squirrel holds on or even progresses. These are dense pine forests Kielder Forest the forests of the Channel Islands and the common factor in all these forests is the presence of a predator the pine marten.

What is remarkable is that the marten does not hunt the two species equally it catches the grey much more easily than the red because the grey is larger spends more time on the ground and is less agile than the red. The result: in areas with martens grey populations are greatly reduced which gives more space back to the reds.

To model this three actor dynamic we use the Lotka-Volterra system write on board

dx/dt = α·x − β·x·y

dy/dt = δ·x·y − γ·y

I note here x(t) the grey population and y(t) the pine marten population α the natural growth rate of the prey β the predation rate measuring the effect of encounters between predators and prey γ the natural mortality rate of the predators δ the efficiency of converting eaten prey into new predators. In the first equation α·x represents the natural reproduction of the greys. The term −β·x·y represents predation: its frequency is proportional to the product of the two populations because the more greys and martens there are the more frequent the encounters. In the second equation δ·x·y is the growth of the martens thanks to the greys consumed. And −γ·y is the natural mortality of the martens. The equilibrium points of the system are: x°= γ/δ and y°= α/β. Around this equilibrium the populations oscillate periodically: the greys increase the martens feed and grow the greys decline the martens decrease due to lack of prey the greys recover. But what interests us here is the effect of the marten on the competition between greys and reds. By reducing the grey population the marten relieves competitive pressure on resources and the "K" of the reds increases. In other words: in forests with martens the parameters of the Volterra model the high predation rate β and the low agility of the grey shift the equilibrium in favour of the red. This is not a coincidence it is a mathematically predictable consequence of the equations.

We started from an English forest and a simple question: why is the red disappearing? And to answer it we built three increasingly rich mathematical tools.
Malthus gives us free growth the idealised starting point. Verhulst adds the reality of limited resources and allows us to quantify the competitive advantage of the grey over the red. Lotka and Volterra go one step further by coupling two species and show how a predator can by lowering the grey population give the red a chance of survival.
What fascinates me in this progression is that the complexity of an ecological phenomenon observed in the field the disappearance of a species across an entire continent emerges from surprisingly simple mathematical rules: a geometric sequence a correction factor two coupled equations.

Thank you for reading also manifesting everything bad on people who say dnr