Effect of DDT | Predator and Prey | BSc Maths Hons Differential Lab Practical Delhi University

Effect of DDT | Predator n Prey

1. PREDATORS-PREY MODEL AND EFFECTS OF DDT Here is where your presentation begins

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3. Introduction and explaination S N Assumptions and Word equations introduction V Word equations explanation R Effect of DDT P Maxima code and Word problems Sameer Shakya Nayan Bhui Viraj Tanti Rahul Kumar Saini Pawan Kumar Members

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6. Predator- Prey model The prey-predator model, also known as the Lotka -Volterra model, is a mathematical representation used in ecology to describe the dynamics between two species: predators and prey. It examines how changes in the population of one species affect the population of the other, and vice versa. Typically, it consists of a pair of coupled differential equations that describe the rates of change of the populations of the two species over time.

7. There are several types of prey-predators interactions like herbivores- which eat plant species. But here we will discuss about interactions of prey and predators and effect of DDTon them. So we will be talking and studying one interesting example of predators-prey interaction occur r ed in the late nineteeth century when the American citrus industry was almost destroyed by the accidental introduction from Australia of cottony cushion scale insect.To combat this pest,its natural predator the Australian ladybird beetle was imported ,but this did not solve the problem and finally DDT was used to kill both predator and prey in abid to eradicate the pest. Surprisingly ,application of DDT to orchards lerc to an increase in the scale insects, the orginal pest,suggesting that the use of pesticide is advantageous to the pest.

8. There are several types of prey-predators interactions like herbivores- which eat plant species. But here we will discuss about interactions of prey and predators and effect of DDTon them. So we will be talking and studying one interesting example of predators-prey interaction occur r ed in the late nineteeth century when the American citrus industry was almost destroyed by the accidental introduction from Australia of cottony cushion scale insect.To combat this pest,its natural predator the Australian ladybird beetle was imported ,but this did not solve the problem and finally DDT was used to kill both predator and prey in abid to eradicate the pest. Surprisingly ,application of DDT to orchards lerc to an increase in the scale insects, the orginal pest,suggesting that the use of pesticide is advantageous to the pest.

9. Hope So till now this shines your head So to further explain this can understand effects of DDT through its various direct and indirect impact on the Preadator-prey interaction . Predator-Prey Background LETS PROCEED TO EFFECTS OF DDT !!

10. Predator-Prey Model (Effect Of DDT) Cascading Effects : The decline of predators and the subsequent increase in prey populations can set off a cascade of effects throughout the ecosystem. For example, an overabundance of prey species may lead to resource depletion, habitat degradation, and competition among prey species themselves. This can further impact other species dependent on the ecosystem, creating a ripple effect of ecological disruption . Feedback Loops: Changes in predator and prey populations can create feedback loops within the ecosystem. For instance, a decline in prey populations due to overconsumption may eventually lead to a decrease in predator populations as their food source dwindles. This can result in a cyclical pattern of population fluctuations, further destabilizing the ecosystem .

11. We make a few preliminary assumptions on whic h to build the model:- 1)Initially we assume the population are large ,sufficiently large to neglect random differences between individuals. 2)We ignore the effect of DDT initially , but modify the model later to in corporate its impact on the system. 3)W e also assume there are only two populations , thw predator and the prey , that affect the ecossystem. 4)We assume that the prey population grows exponentially in the absence of a predator Assumptions for Model 02

12. Compartmental Model There are two two separate quantities:- Number of prey Number of Predators X(t)=Population density of prey Y(t)=Population density of predators B1=Per-capita Birth rate of prey B2=Per-capita Death rate of predators A1=Per-capita Death rate of prey A2=Per-capita Death rate of predators 03

13. Compartmental Model

14. Here we further distinguish between natural prey deaths and prey deaths due to predators. We also distinguish between natural predators births that occurs in the absence of prey and the additional births that would occur due to the predators having more food from eating prey . Compartmental Model

15. Word Equations In the context of ordinary differential equations (ODE), a "word equation" typically refers to a verbal description of a mathematical relationship or process that can be represented by an ODE. It's essentially a description of how one or more variables change over time in terms of other variables or parameters . Word equations are helpful because they provide a conceptual understanding of the relationship between variables before translating them into mathematical notation. They help in formulating the corresponding ODEs to model real-world phenomena accurately. 04

16. Word Equations Now coming further to the equation; { RATE OF CHANGE OF PREY }={ RATE OF NATURAL PREY BIRTHS }-{ RATE OF NATURAL PREY DEATHS }-{ RATE OF PREY KILLED BY PREDATORS } { RATE OF CHANGE OF PREDATORS }={ RATE OF PREDATOR BIRTHS }-{ RATE OF NATURAL PREDATORS DEATH}

17. Word Equations Now coming up to the equation it is assumed that the per-capita rate for the prey is a constant (B1). Also rate of birth for each individual scale insect does not depend on the predator density. Similarly the natural per-capita death rate of scale insect is a constant that is (A1). Also the per capita death rate of prey killed by the predators will depend on predators density this rate (C1Y).

18. Word Equations First look at the constant per capita terms , the prey births and predator deaths. Since the overall rates are the per-capita rates multiplied by the respective population densities ,we write; { Rate of prey Births }=B1*(t) Where B1=( per-capita birth rate of prey ) { Rate of prey natural deaths }=A1*X(t) ; Where A1=( per-capita natural death constant of prey) { Rate of predator deaths }=A2*Y(t) ; Where A2=( per-capita death of predators )

19. Word Equations For the deaths , we denote the per-capita death rate as {C1*Y(t)}.Since its proportional to the predators density with (C1) as the positive constant of proportionality . Thus the rate at which prey are eaten is given by {C1.Y(t).X(t)}. The predator birth rate has a component that is proportional to this rate of prey eaten, so we write; { Rate of Prey killed by predators }={C1.Y(t).X(t)} { Rate of predators Births }={B2*y+ FC1*Y(t).X(t)} ; where F is also a positive constant of proportionality Now ,substitute above equations into word equations we obtain the pair of differential equation

20. Word Equations The differential equations:- DX/DT= B1.X-A1*X-C1*X.Y DY/DT= B2.Y-A2*Y+(F.C1.Y(t)*X(t) We can combine some parameters , lets B1=(b1-A1) and A2=(B2-A2) and C2=F*C1, then we get; DX/DT= B1.X-C1*X.Y DY/DT= B2.Y+F.C1*X*Y-A2.Y Where we assume that B1,A2,C1,C2 are all positive constants.

21. EFFECT OF DDT (ON INTERACTION WITH PREDATOR PREY MODEL If DDT is sprayed on the crops ,we need to modify the equations since this represents a differential mechanism for prey and predator deaths. We assume the per capita death rates due to DDT are contant with different per capita rates p1 & p2 for the prey and pradator respectevily .this means the additional terms p1x(t) for the prey & p2y(t) for the predator ,are….. dx/dt =b1x-c1*xy-p1*x dy /dt =c2*xy-a2*y-p2*y

22. E FFECT OF DDT To investigate the effect of DDT on the two populations .we run the numerical solution again with p1 = p2 =0.1 . The effect of the pesticide on the predator has been to decrease its overall numbers. It appears ,however ,that the mean prey population has increased. This is quite the opposite from the desired effect of the DDT & comes as a result of dealing with a non-linear system. The DDT also reduces the average predator population so that there will be a reduced number of prey deaths due to prey being eaten by the predators.

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26. Insects 02 You can enter a subtitle here if you need it

27. Regions with many insects It’s the biggest planet in the Solar System Despite being red, Mars is a cold place Saturn is a gas giant and has several rings Mars Jupiter Saturn

28. This is a timeline 1xxx Jupiter is the biggest planet of them all Earth is the third planet from the Sun 1xxx 1xxx 2xxx Venus is the second planet from the Sun Mars is actually a very cold place Neptune is very far away from the Sun 2xxx

29. This is an insect Mercury Jupiter is the largest of all the planets It is composed of hydrogen and helium It is the closest planet to the Sun Jupiter Venus Mars Venus is the second planet from the Sun Mars is actually a very cold place It is the farthest planet from the Sun Saturn Neptune

30. Characteristics of an insect Mercury Venus Mars Mercury is the closest planet to the sun and the smallest planet in the Solar System Venus has a beautiful name and is the second closest planet to the Sun Although it is red, Mars is actually one of the coldest planets in the Solar System The Sun is the star at the center of the Solar System Pluto is now considered a dwarf planet Ceres is located in the main asteroid belt Earth Mercury Venus Neptune Jupiter Ceres Neptune is the farthest planet from the Sun.

31. Insect population Follow the link in the graph to modify its data and then paste the new one here. For more info, click here Jupiter is the biggest planet of them all Jupiter Venus Venus is the second planet from the Sun Neptune is very far away from the Sun Neptune

32. How to care for insects Mercury Jupiter is the largest of all the planets It is composed of hydrogen and helium It is the closest planet to the Sun Jupiter Venus Mars Venus is the second planet from the Sun Mars is actually a very cold place It is the farthest planet from the Sun Saturn Neptune 01 02 03 04 05 06

33. Stages of an insect Stage 1 Jupiter is a gas giant and the biggest planet in the Solar System. It's the fourth-brightest object in the night sky Earth is the third planet from the Sun and the only one that harbors life in the Solar System . This is where we all live Despite being red, Mars is actually a cold place . It's full of iron oxide dust, which gives the planet its reddish cast Stage 2 Stage 3 Neptune is the farthest planet from the Sun. It's also the fourth-largest object by diameter in the Solar System Insect

34. This is a conceptual map Insects Neptune Neptune is the farthest planet from the Sun Saturn Saturn is a gas giant and has several rings Venus is the second planet from the Sun Mars is actually a very cold place Venus Mars It’s the closest planet to the Sun Mercury Earth is the third planet from the Sun Earth 03 02 01 04

35. Other analyses Follow the link in the graph to modify its data and then paste the new one here. For more info, click here Jupiter doesn’t have a solid surface Jupiter Venus Venus is the s econd planet from the Sun Neptune is very far away from the Sun Neptune Earth is the third planet from the Sun Earth 25% 35% 45% 85%

36. Questions and answers 01 What is the most important insect ? Earth is the third planet from the Sun Saturn Mars Jupiter Earth Bees 02 Drag the line to the correct answer! Neptune Jupiter was named after a Roman god Despite being red , Mars is very cold Saturn was named after a Roman god 03 04 05 06 Neptune is very far away from us

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