Problem 7
Question
Refer to the negative binomial host-parasitoid model. Problems \(7,8,11\), and 12 are best done with the help of a spreadsheet, but can also be done with a calculator. The negative binomial model is a discrete-generation host- parasitoid model of the form $$ \begin{array}{l} N_{t+1}=b N_{t}\left(1+\frac{a P_{t}}{k}\right)^{-k} \\ P_{t+1}=c N_{t}\left[1-\left(1+\frac{a P_{t}}{k}\right)^{ k}\right] \end{array} $$ for \(t=0,1,2, \ldots\) Evaluate the negative binomial model for the first 10 generations when \(a=0.02, c=3, k=0.75\), and \(b=1.5 .\) For the initial host density, choose \(N_{0}=5\), and for the initial parasitoid density, choose \(P_{0}=0\).
Step-by-Step Solution
Verified Answer
For 10 generations, iterate the provided equations using initial conditions and parameters, preferably in a spreadsheet for efficient computation.
1Step 1: Understand the Model Equations
The negative binomial host-parasitoid model is described by two equations: \( N_{t+1} = b N_{t} \left(1 + \frac{a P_{t}}{k}\right)^{-k} \) and \( P_{t+1} = c N_{t} \left[1 - \left(1 + \frac{a P_{t}}{k}\right)^{ k}\right] \). These equations calculate the next generation's host \(N_{t+1}\) and parasitoid \(P_{t+1}\) populations based on the current populations.
2Step 2: Set Initial Conditions and Parameters
Establish the initial conditions and parameters: \(a = 0.02\), \(c = 3\), \(k = 0.75\), \(b = 1.5\). The initial host density is \(N_0 = 5\) and the initial parasitoid density is \(P_0 = 0\).
3Step 3: Calculate First Generation
Substitute \(N_0 = 5\) and \(P_0 = 0\) into the equations. For generation \(t = 0\), calculate \(N_1\) and \(P_1\): \(N_1 = 1.5 \times 5 \left(1 + \frac{0.02 \times 0}{0.75}\right)^{-0.75} = 7.5\). Since \(P_0 = 0\), \(P_1 = 3 \times 5 \left[1 - \left(1 + \frac{0.02 \times 0}{0.75}\right)^{0.75}\right] = 0\).
4Step 4: Iterate for Subsequent Generations
Repeat the calculation for \(t = 1\) to \(t = 9\), using each previous generation's \(N_t\) and \(P_t\) as input for the next generation. Compute \(N_{t+1}\) and \(P_{t+1}\) using the model equations.
5Step 5: Continue Calculations Until 10 Generations
Continue iterating these calculations for each generation, ensuring to keep track of \(N_t\) and \(P_t\) for all generations from \(t = 0\) to \(t = 9\). This process will eventually give \(N_{10}\) and \(P_{10}\) after the 10th generation.
6Step 6: Use a Spreadsheet for Efficiency
Given the repetitive and computational nature of the problem, it is efficient to use a spreadsheet. Input the equations into a spreadsheet to automate calculations and quickly obtain \(N_t\) and \(P_t\) values for all generations without manual errors.
Key Concepts
Host-Parasitoid ModelPopulation DynamicsDiscrete-Generation Model
Host-Parasitoid Model
The host-parasitoid model is a mathematical framework used to describe the interactions between host species and their parasitoids. In nature, a parasitoid is an organism that spends a significant portion of its life attached to or within a single host organism (which it eventually kills). The host-parasitoid model aims to predict the population dynamics of these two species over time.
In our exercise, the negative binomial host-parasitoid model involves two critical equations:
Each equation represents how the population of hosts \(N\) and parasitoids \(P\) evolves over discrete generations. The terms involve parameters like \(b\), \(a\), \(c\), and \(k\), which can be thought of as controlling the birth rate, the searching efficiency, the conversion efficiency, and the aggregation parameter, respectively. This model assumes that changes in population happen in distinct generational steps rather than continuously.
In our exercise, the negative binomial host-parasitoid model involves two critical equations:
- The equation for the host population:\( N_{t+1} = b N_{t} \left(1 + \frac{a P_{t}}{k}\right)^{-k} \)
- The equation for the parasitoid population:\( P_{t+1} = c N_{t} \left[1 - \left(1 + \frac{a P_{t}}{k}\right)^{k}\right] \)
Each equation represents how the population of hosts \(N\) and parasitoids \(P\) evolves over discrete generations. The terms involve parameters like \(b\), \(a\), \(c\), and \(k\), which can be thought of as controlling the birth rate, the searching efficiency, the conversion efficiency, and the aggregation parameter, respectively. This model assumes that changes in population happen in distinct generational steps rather than continuously.
Population Dynamics
Population dynamics refer to the study of how populations of living organisms change over time and space as a result of birth, death, immigration, and emigration. In the context of the negative binomial model, understanding these dynamics is crucial for predicting how the host and parasitoid populations will fluctuate from one generation to the next.
In our specific model, for each generation:
In our specific model, for each generation:
- The host population \(N\) increases by births, which is influenced by the parameter \(b\), representing the reproductive rate of the hosts.
- The parasitoid population \(P\) grows through its ability to convert hosts into new parasitoids, modulated by the parameter \(c\).
Discrete-Generation Model
A discrete-generation model focuses on populations where each generation is distinct, without overlapping lifecycles between old and new generations. This concept fits the host-parasitoid interactions well, as they often have distinct life stages. Each cycle of the model simulates the complete lifecycle of both species:
In the negative binomial version of the host-parasitoid model, generations do not overlap, meaning that individuals born in one generation do not directly interact with those born in another. This helps simplify the calculations by focusing on one set of hosts and parasitoids at a time, using initial conditions to predict subsequent generations.
Without overlapping generations, this model can effectively study seasonal species or those with clear-cut generational boundaries. This approach is particularly useful in ecological studies where clarity on sequential events is crucial, such as understanding pest control dynamics, and managing population stability in conservation efforts.
- From birth to dying off for hosts.
- From emergence to impacting the host population for parasitoids.
In the negative binomial version of the host-parasitoid model, generations do not overlap, meaning that individuals born in one generation do not directly interact with those born in another. This helps simplify the calculations by focusing on one set of hosts and parasitoids at a time, using initial conditions to predict subsequent generations.
Without overlapping generations, this model can effectively study seasonal species or those with clear-cut generational boundaries. This approach is particularly useful in ecological studies where clarity on sequential events is crucial, such as understanding pest control dynamics, and managing population stability in conservation efforts.
Other exercises in this chapter
Problem 6
In Problems 1-14, use the properties of limits to calculate the following limits: $$ \lim _{(x, y) \rightarrow(-5,1)} y\left(x y+x^{2} y^{2}\right) $$
View solution Problem 7
Find \(\frac{d z}{d t}\) for \(z=f(x, y)\) with \(x=u(t)\) and \(y=v(t)\).
View solution Problem 7
Evaluate each function at the given point. (a) \(f_{1}(x, y)=2 x-3 y^{2}\) at \((-1,2)\) (b) \(f_{2}(y, x)=2 x-3 y^{2}\) at \((-1,2)\)
View solution Problem 7
The tangent plane at the indicated point \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. $$ f(x, y)=e^{x^{2}+y^{2}} ;(1,0, e) $$
View solution