When an advantageous gene appears in a population, or when an infectious person appears in a population of susceptibles, a traveling wave can often appear. In the first case, the advantageous gene can spread through the population, while in the second case it is the infection that spread

a. First, consider the case of the advantageous gene. AssumActually, to be honest, this equation is not quite the one that is usually used to model the spread of an advantageous gene, which is often modelled by Fisher’s equation. However, the behaviour of this equation is very similar to the behaviour of solutions of Fisher’s equation.
ing that the gene appears first at the origin (just to make things simpler), the spread of the gene through the population can be approximately described by an equation of the form P(x, y, t) = 1 1 + e r 2−t , where r 2 = x 2 + y 2 (so r is the distance from the origin)
and t is time. P is the proportion of the advantageous gene in the population.
i. Plot P for various increasing values of t.
ii. Show that the equation for P describes a spreading wave front of the gene, leaving a high probability behind the wave. Why could this model the spread of an advantageous gene?

b. Next, consider a similar problem, the spread of an infection through a susceptible population. However, now we suppose that people recover from the infection. In this case, the function P from part

a. of this question won’t work.
(Why not?) Instead, we model the spread of a non-lethal infection by the equation Again, we’re cheating a bit, as this equation would not typically be used to model the spread of an infection, which tends to be a much more complicated process. However, the overall shape of this equation is qualitatively correct.
I(x, y, t) = 1 −
(r 2 − t)
4 1 + (r 2 − t)
4 , where, as before, r 2 = x 2 + y 2 and t is time. I is the proportion of the population that is infected. Plot I for various values of t and explain why this could describe the spread of an infection, from which people recover.

a. First, consider the case of the advantageous gene. AssumActually, to be honest, this equation is not quite the one that is usually used to model the spread of an advantageous gene, which is often modelled by Fisher’s equation. However, the behaviour of this equation is very similar to the behaviour of solutions of Fisher’s equation.
ing that the gene appears first at the origin (just to make things simpler), the spread of the gene through the population can be approximately described by an equation of the form P(x, y, t) = 1 1 + e r 2−t , where r 2 = x 2 + y 2 (so r is the distance from the origin)
and t is time. P is the proportion of the advantageous gene in the population.
i. Plot P for various increasing values of t.
ii. Show that the equation for P describes a spreading wave front of the gene, leaving a high probability behind the wave. Why could this model the spread of an advantageous gene?

b. Next, consider a similar problem, the spread of an infection through a susceptible population. However, now we suppose that people recover from the infection. In this case, the function P from part

a. of this question won’t work.
(Why not?) Instead, we model the spread of a non-lethal infection by the equation Again, we’re cheating a bit, as this equation would not typically be used to model the spread of an infection, which tends to be a much more complicated process. However, the overall shape of this equation is qualitatively correct.
I(x, y, t) = 1 −
(r 2 − t)
4 1 + (r 2 − t)
4 , where, as before, r 2 = x 2 + y 2 and t is time. I is the proportion of the population that is infected. Plot I for various values of t and explain why this could describe the spread of an infection, from which people recover.

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