Puffballs fungi grow until they become surprinsingly fragile. They develop an aperture, dry out, or become...

Transcribed Image Text:

Puffballs fungi grow until they become surprinsingly fragile. They develop an aperture, dry out, or become brittle so that a cloud of spores can be ejected in bursts if they are touched. The mass M of a Puffball fungus grows logistically: it first increases exponentially and then plateaus. The evolution of the mass M of this fungus can be modelled using the following ODE dM/dt-rM(1-M/K) a) Given a growth rate r=0.2 grams per week and a carying capacity (or here maximal mass) K=10 grams, what are the fixed points of this system? Do the fixed points depend on growth rate? b) Assume a starting mass of M(O)=1 gram. Use the Euler method to solve and plot the evolution of the mass M(t) as a function of time over a period of 100 weeks with dt-0.1 weeks. Your plot should have a title, axis labels, and plotted using yellow dots. c) On the 61st week, an animal passes by and provokes the expulsion of spores. When this happens, the fungus loses 80% of its mass, and resumes its growth normally afterwards. Factor this 80% loss into your Euler's equation from part b) and plot the assoicated solution. d) Based on your answer in c), at what time(s) will the mass of the fungus be 270% of its starting mass? Puffballs fungi grow until they become surprinsingly fragile. They develop an aperture, dry out, or become brittle so that a cloud of spores can be ejected in bursts if they are touched. The mass M of a Puffball fungus grows logistically: it first increases exponentially and then plateaus. The evolution of the mass M of this fungus can be modelled using the following ODE dM/dt-rM(1-M/K) a) Given a growth rate r=0.2 grams per week and a carying capacity (or here maximal mass) K=10 grams, what are the fixed points of this system? Do the fixed points depend on growth rate? b) Assume a starting mass of M(O)=1 gram. Use the Euler method to solve and plot the evolution of the mass M(t) as a function of time over a period of 100 weeks with dt-0.1 weeks. Your plot should have a title, axis labels, and plotted using yellow dots. c) On the 61st week, an animal passes by and provokes the expulsion of spores. When this happens, the fungus loses 80% of its mass, and resumes its growth normally afterwards. Factor this 80% loss into your Euler's equation from part b) and plot the assoicated solution. d) Based on your answer in c), at what time(s) will the mass of the fungus be 270% of its starting mass?

Answer rating: 100% (QA)

a To find the fixed points of the system we need to solve for M when dMdt 0 The equation is dMdt rM1 MK 0 This equation has two solutions M 0 1 MK 0 w... View the full answer