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Modeling the Dynamics of Life Calculus and Probability for Life Scientists 3rd edition Frederick R. Adler - Solutions
Population 4 starting from 50 individuals.The following table gives the immigration (positive values) or emigration (negative values) for four populations over a period of 10 years. Find the population after 10 years starting from the given initial population, and check whether the population has
What is the per capita production of population 1 during these years?The following table describes populations that are growing through production. For each, sketch a graph, find the per capita production in each year, and describe the growth of the population in words. When will the population
What is the per capita production of population 2 during these years?The following table describes populations that are growing through production. For each, sketch a graph, find the per capita production in each year, and describe the growth of the population in words. When will the population
What is the per capita production of population 3 during these years?The following table describes populations that are growing through production. For each, sketch a graph, find the per capita production in each year, and describe the growth of the population in words. When will the population
A population starts at size 1000 and grows with per capita production rt that alternates between 0.7 and 1.5. In each of the above populations, per capita production changes over time according to a fixed pattern (rather than a random pattern). For each population, find the population during the
What is the per capita production of population 4 during these years?The following table describes populations that are growing through production. For each, sketch a graph, find the per capita production in each year, and describe the growth of the population in words. When will the population
What is the change in population 1 during these years?The following table describes populations that are growing through immigration or emigration. For each, sketch a graph, find the number of individuals that arrived or left in each year, and describe the growth of the population in words. When
What is the change in population 2 during these years?The following table describes populations that are growing through immigration or emigration. For each, sketch a graph, find the number of individuals that arrived or left in each year, and describe the growth of the population in words. When
What is the change in population 3 during these years?The following table describes populations that are growing through immigration or emigration. For each, sketch a graph, find the number of individuals that arrived or left in each year, and describe the growth of the population in words. When
What is the change in population 4 during these years?The following table describes populations that are growing through immigration or emigration. For each, sketch a graph, find the number of individuals that arrived or left in each year, and describe the growth of the population in words. When
On island 1.Suppose the states of populations on four islands are described in the following table.For each, illustrate what is happening with a graph, and describe it in words. Does any of the islands have a pattern that can be described deterministically?
On island 2.Suppose the states of populations on four islands are described in the following table.For each, illustrate what is happening with a graph, and describe it in words. Does any of the islands have a pattern that can be described deterministically?
On island 3.Suppose the states of populations on four islands are described in the following table.For each, illustrate what is happening with a graph, and describe it in words. Does any of the islands have a pattern that can be described deterministically?
A population starts at size 1000 and grows with per capita production rt that has a 3 year cycle, first 0.7, then 0.9, and then 1.6. In each of the above populations, per capita production changes over time according to a fixed pattern (rather than a random pattern). For each population, find the
On island 4.Suppose the states of populations on four islands are described in the following table.For each, illustrate what is happening with a graph, and describe it in words. Does any of the islands have a pattern that can be described deterministically?
The simple models of stochasticity described in the book leave out a lot of biological detail. Use your imagination to add some of that detail back.1. Think of two biological factors that are neglected in the stochastic model bt = rtbt, Equation 6.1.1.2. Think of three factors neglected in the
A population starts at size 1000 and grows with per capita production rt that has a 3 year cycle, first 0.7, then 0.9, and then 1.5. In each of the above populations, per capita production changes over time according to a fixed pattern (rather than a random pattern). For each population, find the
A population starts at size 100, and receives 10 immigrants in the first year, loses 5 emigrants in the second year, receives 10 immigrants in the third year, and so forth. In each of the above population, immigration changes over time according to a fixed pattern (rather than a random pattern).
A population starts at size 100, and receives 10 immigrants in the first year, loses 12 emigrants in the second year, receives 10 immigrants in the third year, and so forth. In each of the above population, immigration changes over time according to a fixed pattern (rather than a random pattern).
A population starts at size 50, and receives 10 immigrants in the first year, loses 12 emigrants in the second year, and gains 5 immigrants in the third year. It then repeats this 3 year cycle. In each of the above population, immigration changes over time according to a fixed pattern (rather than
A population starts at size 20, and has a 6 year cycle: gain 1, lose 2, gain 3, lose 4, gain 3, lose 2. In each of the above population, immigration changes over time according to a fixed pattern (rather than a random pattern). For each population, find the population in each of the first 6 years
Population 1 starting from 100 individuals.The following table gives the per capita production for four populations over a period of 10 years. Find the population over the 10 years starting from the given initial population, sketch a graph, and check whether the population has increased or
The probability it leaves is 0.3 each second.
Each second, 3% of the chemical in container 1 enters container 2, and 10% of the chemical in container 2 returns to container 1 (compare with Exercise 4). In many ways, probabilities act like fluids. For each of the following models of chemical exchange, let ct represent the amount in container 1
The fraction of grand-offspring (second generation) with genotype AA. Compute the above probability for a selfing plant using Figure 6.2.9.
The fraction of third-generation offspring with genotype AA. Compute the above probability for a selfing plant using Figure 6.2.9.
The fraction of fourth-generation offspring with genotype AA. Compute the above probability for a selfing plant using Figure 6.2.9.
All of the homozygous offspring survive, and half of the heterozygous offspring survive. Suppose that the fraction of homozygous and heterozygous offspring that survive self-fertilization by a heterozygote is measured. Find the fraction of surviving offspring that are heterozygous in the following
Half of the homozygous offspring survive, and one third of the heterozygous offspring survive. Suppose that the fraction of homozygous and heterozygous offspring that survive self-fertilization by a heterozygote is measured. Find the fraction of surviving offspring that are heterozygous in the
What are all the possible matings? What would be the heights of the offspring? What is the probability that a 40-cm tall plant mates with a 40-cm tall plant? Consider the above case of blending inheritance. A population of plants starts out with an equal number of individuals of heights 40 and 60
Suppose that these offspring now mate with each other. Find all possible matings and the resulting offspring heights. Consider the above case of blending inheritance. A population of plants starts out with an equal number of individuals of heights 40 and 60 cm. The parents mate randomly, and the
The probability it leaves is 0.03 each second.
Find the probability of each of the possible matings of the offspring. Out of 100 plants, about how many would have height 50 cm? Consider the above case of blending inheritance. A population of plants starts out with an equal number of individuals of heights 40 and 60 cm. The parents mate
With blending inheritance, the height of the offspring is equal to the average height of the four grandparents. Find the probability that all four grandparents have height 40 cm and thus the probability that a plant in the second generation has height 40 cm. Consider the above case of blending
Find all the ways that the heights of the grandparents average to exactly 50 cm. Consider the above case of blending inheritance. A population of plants starts out with an equal number of individuals of heights 40 and 60 cm. The parents mate randomly, and the height of an offspring is exactly equal
A certain highly mutable gene has a 1.0% chance of mutating each time a cell divides. Suppose that there are 15 cell divisions between each pair of generations. What is the chance that the gene mutates in one generation, during the course of those 15 cell divisions? If there were 100 such genes,
A herd of lemmings is standing at the top of a cliff. Each jumps off with probability 0.2 each hour. What is the probability that a particular lemming remains on top of the cliff after 3 hours? If 5000 lemmings are standing around on top of the cliff, about how many will remain after 3 hours?
A molecule has a 5.0% chance of binding to an enzyme each second and remains permanently attached thereafter. If the molecule starts out unbound, find the probability that it is bound after 10 seconds. How long would it take for the molecule to have bound with probability 0.95?
In tropical regions, caterpillars suffer extremely high parasitism, sometimes as high as 15% per day. In other words, a caterpillar is attacked by a parasitoid with probability 0.15 each day. If a caterpillar takes 25 days to develop, what is the probability it survives? If a female lays 50 eggs,
Suppose that each mutant gene in Exercise 23 has a 1.0% chance of mutating back to the original type each cell division. Use the Markov chain approach to find the fraction of mutant genes after 15 cell divisions. How much difference does the correction mechanism make?
Suppose that the lemmings in Exercise 24 can sometimes crawl back up the cliff. In particular, suppose that a lemming at the bottom of the cliff climbs back up with probability 0.1 each hour. What is the probability that a particular lemming is on top of the cliff after 3 hours? If 5000 lemmings
Suppose that bound molecules in Exercise 25 have a 2.0% chance of unbinding from the enzyme each second. Find the fraction of molecules that are bound in the long run. What is the probability that a molecule is bound after 10 seconds?
The probability it leaves is 0.3 each second, and the probability it returns is 0.2 each second. The following probabilities describe molecules that can hop into and out of a cell. For each, find a discrete-time dynamical system for the probability that the molecule is inside. Find the probability
Suppose that the caterpillars in Exercise 26 have some chance of eliminating their attacker, thus becoming a caterpillar again. In particular, suppose that a caterpillar has a 0.03 chance of eliminating a parasitoid each day. Find the probability that a caterpillar is a caterpillar after 25 days.
The probability that an offspring is heterozygous is 0.6.
The probability that an offspring is heterozygous is 0.4.
The probability that an offspring is heterozygous is 0.2.
The probability that an offspring is heterozygous is 0.9.
A heterozygous plant with genotype Aa self-pollinates. Find the probability that an offspring is tall for the following genetic systems.1. Only plants that have two A alleles are tall (the allele A is recessive).2. Plants that have either one or two A alleles are tall (the allele A is dominant).3.
A heterozygous plant with genotype Aa self-pollinates, and then its offspring also self-pollinate. Find the probability that the offspring of the offspring are tall for the following genetics systems.1. Only plants that have two A alleles are tall.2. Plants that has either one or two A alleles are
The probability it leaves is 0.03 each second, and the probability it returns is 0.1 each second. The following probabilities describe molecules that can hop into and out of a cell. For each, find a discrete-time dynamical system for the probability that the molecule is inside. Find the probability
What is the genotype at the eye color locus in the first generation? Often geneticists want to change one allele in an out-crossing organism while keeping the rest of the genome the same. For example, they might wish to take a specially designed stock of flies and alter the eye color from red to
What is the genotype at the eye color locus in the second and subsequent generations? Often geneticists want to change one allele in an out-crossing organism while keeping the rest of the genome the same. For example, they might wish to take a specially designed stock of flies and alter the eye
What fraction of flies will have the a allele (at the second locus) after t generations? Often geneticists want to change one allele in an out-crossing organism while keeping the rest of the genome the same. For example, they might wish to take a specially designed stock of flies and alter the eye
How many back-crosses would be necessary to purge 99.9999% of the inferior genes from the white-eyed fly? Often geneticists want to change one allele in an out-crossing organism while keeping the rest of the genome the same. For example, they might wish to take a specially designed stock of flies
Suppose meiotic drive affects the pollen only and that 80% of the pollen grains from a heterozygote carry the A allele. Ovules are normal, and 50% of them carry the A allele. What fraction of offspring from a selfing heterozygote will be heterozygous? One force that can alter the ratio of
The molecule in Exercise 1. Draw cobweb diagrams based on the discrete-time dynamical systems in the earlier exercise.
Suppose meiotic drive affects both pollen and ovules and that 80% of the pollen grains and ovules from a heterozygote carry the A allele. What fraction of offspring from a selfing heterozygote will be heterozygous? One force that can alter the ratio of heterozygotes produced by a selfing
Suppose meiotic drive affects both pollen and ovules but that 80% of the pollen grains carry the A allele while 80% of ovules carry the a allele. What fraction of offspring from a selfing heterozygote will be heterozygous? One force that can alter the ratio of heterozygotes produced by a selfing
Heterozygosity in inbreeding organisms can be restored by mutation. Suppose that mutations always create brand new alleles. Suppose that each parental allele has a probability 0.01 of mutating.1. Suppose first that the parent has genotype AA. What is the probability that the allele that came from
The molecule in Exercise 2. Draw cobweb diagrams based on the discrete-time dynamical systems in the earlier exercise.
The molecule in Exercise 3. Draw cobweb diagrams based on the discrete-time dynamical systems in the earlier exercise.
The molecule in Exercise 4. Draw cobweb diagrams based on the discrete-time dynamical systems in the earlier exercise.
Each second, 30% of the chemical in container 1 enters container 2, and 20% of the chemical in container 2 returns to container 1 (compare with Exercise 3). In many ways, probabilities act like fluids. For each of the following models of chemical exchange, let ct represent the amount in container 1
For the given sets A and B, find A ⋂ B, A ⋃ B, and Ac (the complement of A). 1. A and B are subsets of the set S = {0, 1, 2, 3, 4}. A = {0, 1, 2} and B = {0, 2, 4}. 2. A and B are subsets of the set S = {0, 1, 2, 3, 4, 5}. A = {0, 1, 2} and B = {0, 2, 4, 5). 3. A and B are subsets of the set of
A and B disjoint, B and C not disjoint, A and C not disjoint. Draw Venn diagrams with sets A, B, and C satisfying the above requirements.
No two sets disjoint, but A⋂B⋂C empty. Draw Venn diagrams with sets A, B, and C satisfying the above requirements.
No two sets disjoint, and A⋂B⋂C nonempty. Draw Venn diagrams with sets A, B, and C satisfying the above requirements.
The sets A and B in Exercise 5.The following formula gives the probability of the union of any two events, whether or not they are disjoint,Pr(A ‹ƒ B) = Pr(A) + Pr(B) - Pr(A ‹‚ B).As indicated in the figure, adding the area in A and B counts the area in the intersection A
The sets A and B in Exercise 6.The following formula gives the probability of the union of any two events, whether or not they are disjoint,Pr(A ‹ƒ B) = Pr(A) + Pr(B) - Pr(A ‹‚ B).As indicated in the figure, adding the area in A and B counts the area in the intersection A
Using the probabilities in Exercise 5, check the formula on the sets C = {l, 2, 3} and D = {0, 1, 2}.The following formula gives the probability of the union of any two events, whether or not they are disjoint,Pr(A ‹ƒ B) = Pr(A) + Pr(B) - Pr(A ‹‚ B).As indicated in the figure,
Using the probabilities in Exercise 6, check the formula on the sets C = (2, 3, 4} and D = {0, 1, 2}.The following formula gives the probability of the union of any two events, whether or not they are disjoint,Pr(A ‹ƒ B) = Pr(A) + Pr(B) - Pr(A ‹‚ B).As indicated in the figure,
Give the sample spaces associated with the following experiments. Say how many simple events there are and list them if there are fewer than ten. If there are more than ten, list three simple events.1. We cross two plants with genotype bB and check the genotype of one offspring.2. We cross two
Give the sample spaces associated with the following experiments. Say how many simple events there are and list them if there are fewer than ten. If there are more than ten, list three simple events.1. Two molecules jump in and out of a cell. We record how many molecules are inside at times 1 and
We start 100 molecules in a cell and count the number, N, that remain after 10 minutes. Give five simple events that are included in the following events.1. N < 10.2. N > 90.3. N is odd.4. 30 ≤ N ≤ 32 or 68 ≤ N ≤ 70.
We start 100 molecules in a cell and count the number, N, that remain after 10 minutes. Find the union and intersection of the following events.1. Event A is N < 10, and event B is N > 5.2. Event A is N > 10, and event B is N < 5.3. Event A is N > 10, and event B is N > 5.4. Event
All molecules left before minute 5. We follow four individually labeled molecules and record the minute ti when molecule I leaves the cell. For example, if ti = 1, t2 = 3, t3 = 6, and t4 = 2, the first molecule left during minute 1, the second left during minute 3, the third left during minute 6,
Molecules 1, 2, and 4 left before minute 5, and molecule 3 left after minute 7. We follow four individually labeled molecules and record the minute ti when molecule I leaves the cell. For example, if ti = 1, t2 = 3, t3 = 6, and t4 = 2, the first molecule left during minute 1, the second left during
All odd-numbered molecules left at odd times. We follow four individually labeled molecules and record the minute ti when molecule I leaves the cell. For example, if ti = 1, t2 = 3, t3 = 6, and t4 = 2, the first molecule left during minute 1, the second left during minute 3, the third left during
All odd-numbered molecules left at odd times, and all even-numbered molecules left at even times. We follow four individually labeled molecules and record the minute ti when molecule I leaves the cell. For example, if ti = 1, t2 = 3, t3 = 6, and t4 = 2, the first molecule left during minute 1, the
Give two mathematically consistent ways of assigning probabilities to the results of the following experiments. Try to make one of your assignments biologically reasonable.1. The situation in Exercise 17.2. The situation in Exercise 18.
Give two assignments, different from those in the text, of probabilities when counting the number of molecules inside a cell starting from an initial number of 3. Compute Pr(N is odd) and Pr(N ≠ 1) in each case.1. Create an assignment where Pr(N = 1) is larger than the probability of any other
The sample space is S = {0, 1, 2, 3, 4). Suppose that Pr({0}) = 0.2 Pr({1}) = 0.3 Pr({2}) = 0.4 Pr({3}) = 0.1 Pr({4}) = 0.0. Find Pr(A) and Pr(Ac) if A = {0, 1, 2} and Pr(B) if B = {0, 2, 4}. Is Pr(A ⋃ B) = Pr(A) + Pr(B)? Why or why not? For the given sets and sample spaces, show that the
The sample space is S = (0, 1, 2, 3, 4}. Suppose that Pr({0}) = 0.1 Pr({1}) = 0.3 Pr({2}) = 0.4 Pr({3}) = 0.1 Pr({4}) = 0.1. Find Pr(A) and Pr(Ac) if A = {0, 2} and Pr(B) if B = {3, 4}. Is Pr(A ⋃ B) = Pr(A) + Pr(B)? Why or why not? For the given sets and sample spaces, show that the assignment of
The sample space is S = {0, 1, 2, 3, 4}. Suppose that Pr({0}) = 0.2, Pr({1}) = 0.1, Pr({2}) = 0.4, and Pr({3}) = 0.1. Find Pr(A) and Pr(Ac) if A = {4} and Pr(B) if B = {3, 4}.For the given sets and sample spaces, show that the assignment of probabilities is mathematically consistent and use them to
The sample space is S = {0, 1, 2, 3, 4). Suppose that Pr({0}) = 0.3, Pr({l}) = 0.2, Pr({2}) = 0.4, and Pr({4}) = 0.1. Find Pr({3}), Pr({1, 2, 3}), and Pr({2, 4}). For the given sets and sample spaces, show that the assignment of probabilities is mathematically consistent and use them to compute the
A and B disjoint, B and C disjoint, A and C not disjoint. Draw Venn diagrams with sets A, B, and C satisfying the above requirements.
For the given sample spaces, find a set of mutually exclusive and collective exhaustive events with the given number of elements. 1. S = {0, 1, 2, 3, 4}. Find a set of two mutually exclusive and collective exhaustive events. 2. S = {0, 1, 2, 3, 4}. Find a set of three mutually exclusive and
The probability that the total on the two die is 5 or more. Somebody invents a three-sided die that gives scores of 1, 2, or 3, each with probability 1/3. Two such die are rolled. Use the law of total probability to find the above probability, and then find them directly by counting.
The probability that the total on the two die is odd. Somebody invents a three-sided die that gives scores of 1, 2, or 3, each with probability 1/3. Two such die are rolled. Use the law of total probability to find the above probability, and then find them directly by counting.
The probability that the second roll was larger than the first. Somebody invents a three-sided die that gives scores of 1, 2, or 3, each with probability 1/3. Two such die are rolled. Use the law of total probability to find the above probability, and then find them directly by counting.
Find the probability that the first roll is a 3 if the total of the two rolls is greater than or equal to 4 (based on Exercise 9). Consider again the three-sided die that gives a score of 1 with probability 1/3, a score of 2 with probability 1/3, and a score of 3 with probability 1/3. Two such die
Find the probability that the first roll is a 3 if the total of the two rolls is greater than or equal to 5 (based on Exercise 10). Consider again the three-sided die that gives a score of 1 with probability 1/3, a score of 2 with probability 1/3, and a score of 3 with probability 1/3. Two such die
Find the probability that the first roll is a 3 if the total of the two rolls is odd. Consider again the three-sided die that gives a score of 1 with probability 1/3, a score of 2 with probability 1/3, and a score of 3 with probability 1/3. Two such die are rolled. Use Bayes' theorem to find the
Find the probability that the first roll is a 1 if the second roll is greater than the first. Consider again the three-sided die that gives a score of 1 with probability 1/3, a score of 2 with probability 1/3, and a score of 3 with probability 1/3. Two such die are rolled. Use Bayes' theorem to
Four balls are placed in a jar, two red, one blue, and one yellow. Two are removed at random.1. You are told that the first ball removed was red. What is the probability that the second is red?2. You are told that at least one of the two removed is red. What is the probability that both are red?3.
Give a set of three mutually exclusive and collectively exhaustive sets for each of the following sample spaces.1. The situation in Exercise 17.2. The situation in Exercise 18.3. The situation in Exercise 19.4. The situation in Exercise 20.5. The situation in Exercise 23.6. The situation in
She sees an eagle with probability 0.2 during an hour of observation, a jackrabbit with probability 0.5, and both with probability 0.05.An ecologist is looking for the effects of eagle predation on the behavior of jackrabbits. In each of the following cases,a. Draw a Venn diagram to illustrate the
She sees an eagle with probability 0.2 during an hour of observation, a jackrabbit with probability 0.5, and both with probability 0.15.An ecologist is looking for the effects of eagle predation on the behavior of jackrabbits. In each of the following cases,a. Draw a Venn diagram to illustrate the
If 30% of the cells are young, what is the probability that a cell stains properly? A lab is attempting to stain many cells. Young cells stain properly 90% of the time and old cells stain properly 70% of the time.
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