Solve problems relating to logistic models

Twin islands Futabaa and Futhu follows the same language. A new word is introduced in the their language by Futabaa Language Research Science Center. The predicted rate of adoption in Futabaa population for the new word is 0.45. The population of Futabaa is 2 million and Futhu is 8 million. After 10 days, the word starts covering Futhu population. The overall adoption rate predicted is 0.68. If the initial popoulation using the new word on Futabaa island is 20,000. Calculate the time in days when 95% of combined population of both islands start using the new word

A forest on foot hills has population of snakes and rats. The snake is natural predator of rats and the population growth of both follows the predator-prey equations given by as,

Rat :

Snake :

The population growth is in equilibrium and oscillating periodically over time, represented graphically below,

In equilibrium, what is frequency of oscillation for the population growth.

The local pond is filled with phytoplankton with squids as their natural predators to maintain the population growth of phytoplankton. The Lotka-Volterra equation describes predator prey relationship as:

Phytoplankton :

Squids :

The rate of population growth of phytoplankton and squids is given by above equation with ‘a’ is the growth rate of phytoplankton and ‘b’ is predation rate. Factor ‘c’ represents the mortality rate of squids and ‘d’ as growth rate of squids with predation.

Industrial effluent is being discharged into the lake and is toxic to both species, but directly consumed by phytoplankton only. If squids start consuming toxic borne plankton, the population growth rate will change. What pair of modified equations describe this scenario? Consider ‘x’ as toxicity rate common to both species.

 If two predators with one prey occurs in natural habitat, the Lotka Volterra equation for the population growth is given as

The ‘a’ is growth rate of prey, ‘p1’ is predation rate of predator 1 and ‘p2’ is predation rate of predator 2. ‘b’ is the death rate of predator 1 and ‘c’ is the death rate of predator 2. ‘d1’ and ‘d2’ are the growth rate of both predators with predation. If death rate of predator 1 is greater than predator 2 (b>c), then which species will go extinct as time tends to infinity (t→∞)?

The logistic function in form of differential equation is given by,

where, P is population growth as function of time, ‘r’ is the growth rate and K is the carrying capacity.

If the carrying capacity is the also function of time K(t) and changing with time given as ,

How the population growth will follow the varying carrying capacity?

 In a natural habitat, wolf and rabbit form a predator prey relationship and the population growth of both species is governed by Lotka Volterra equation. The relation is given as,

Rabbit :

Wolf :

where, a is growth rate of rabbit population

b is predation rate

c is the death rate of wolf population

d is growth rate due to predation

As observed in the habitat, single wolf consumes multiple rabbits in a day. What condition must the equations satisfy to reflect this observation?

In the predator prey model if the population grows logistically, which pair of Lotka Volterra equations describe the population growth of prey and predator? The carrying capacity is given by ‘K’

In autocatalysis reaction, the product of reaction act as auto catalyst to convert reactant into product. The reaction is triggered with very minimal addition of product.

X → Y

The autocataytic reaction follows the Lotka Volterra equation pair representing the concentration of reactant (X) and product (Y) over time. Which linear equation represents the concentration variation of product with concentration changes in reactant? [X] is concentration of reactant and [Y] is concentration of product.

The Lotka-Volterra equation represent the growth of population in a predator prey relationship. The Lotka-Volterra is given as:

In a natural habitat, if the population growth is in equilibrium at {y=0 and x=0}, which species will go extinct?

The population of pig in slaughter farm grows logistically at rate of ‘r’ and shrinks due slaughter at kill rate ‘k’. If the owner of farm wants to keep the population at 70% of capacity, what must value of  k/r  to satisfy the given condition?

The Bluetooth headset manufacturing unit has launched the new model and it’s a hit in the market. The sales team has predicted a logistical growth in sale of new headset with rate ‘0.5’ ,initial sale of 50000 sets and sale saturation around 1 million pieces. In order to maintain the demand surge, the production employees have to work overtime. If the company has to pay $50 per hour of overtime, calculate the amount overtime expenses company has to bear with 50 employees in production department and overtime of 4 hours per day.

The growth of ant population in an farm follows the standard logistic growth model. The growth rate is defined as ‘r’ with carrying capacity as ‘K’. As per study, the ant population only grows if the initial population is greater than a threshold number of ants ‘T’. Which equation describe the growth of ant population in the farm per unit time?

A person is suffering from brain tumor which is growing logistically. The pain is unbearable if the growth is at high rate and bearable if growth slows down. The initial size of tumor detected is 4 units and doctor has predicted it will grow upto 15 units in days. For how many days, the person has to suffer unbearable pain? (The standard growth rate of any maliginate tumor as per medical science is 0.23) . Round off the answer to upper value.

The population of shrimp in a pond is greater the carrying capacity of shrimp population. The initial population of shrimp is 1500 and carrying capacity is 900, calculate rate of decline if the population falls to  just 1% higher than carrying capacity in 6 months.

A mobile application named “Eapa” is rolled on google play store as contactless payment mode. Only 30% of population owns a smart phone to download the application. The total population of the country is 10 million. The developers are expecting a download rate of 0.1 with initial download of 20,000. (Assume every smartphone owner downloads the application). Calculate the number of days required to hit 1 million downloads.