Problem 62 - Algebra and Trigonometry with Analytic Geometry

Question

Cattle population A rancher has 750 head of cattle consisting of 400 adults (aged 2 or more years), 150 yearlings, and 200 calves. The following information is known about this particular species. Each spring an adult female gives birth to a single calf, and \(75 \%\) of these calves will survive the first year. The yearly survival percentages for yearlings and adults are \(80 \%\) and \(90 \%\), respectively. The male-female ratio is one in all age classes. Estimate the population of each age class (a) next spring (b) last spring

Step-by-Step Solution

Answer

(a) Next spring: 680 cattle. (b) Last spring: 1094 cattle.

Step 1: Number of Female Adults

Calculate the number of female adults. Since the male-female ratio is one, half of the 400 adults are female. Number of female adults = \( \frac{400}{2} = 200 \)

Step 2: New Calves Next Spring

Each female adult gives birth to one calf. So, the number of calves born next spring is equal to the number of female adults. Number of new calves = 200

Step 3: Surviving Calves to Yearlings

Only 75% of calves survive to become yearlings. Calculate the number of surviving calves.Surviving calves = \(200 \times 0.75 = 150\)

Step 4: Yearlings to Adults

80% of this year's yearlings survive to become adults next year.Surviving yearlings = \(150 \times 0.8 = 120\)

Step 5: Surviving Adults

90% of adults survive to the next year.Surviving adults = \(400 \times 0.9 = 360\)

Step 6: Total Population Next Spring

Combine the surviving adults, the surviving yearlings, and the newly born calves to estimate the population next spring. Total adults = 360 Total yearlings = 120 Total calves = 200 Total population = 360 + 120 + 200 = 680

Step 7: Reconstruct Last Year's Calf Population

To estimate last spring's calf population, assume current yearlings were calves last spring. Back-calculate using those numbers.Last year's calves = \(\frac{150}{0.75} = 200\)

Step 8: Reconstruct Last Year's Yearling Population

Assuming the current adults were yearlings last year, calculate the number of yearlings from last spring.Last year's yearlings = \(\frac{360}{0.8} = 450\)

Step 9: Previous Adult Population

Calculate last spring's adult population using the number of adults that survived to this spring.Last spring's adults = \(\frac{400}{0.9} = 444.44\), approximated to 444

Step 10: Total Population Last Spring

Sum the reconstructed populations for calves, yearlings, and adults from last spring. Total last spring = 200 calves + 450 yearlings + 444 adults = 1094

Key Concepts

Cattle Population Dynamics

Understanding cattle population dynamics involves looking at how a cattle population changes over time. Key factors that drive these dynamics include birth rates, death rates, and the transition of individuals between different age classes. In our scenario, each adult cow contributes to the population by giving birth to a calf each year. This process increases the calf population at a predictable rate, as long as each adult female is healthy and fertile.

However, not all calves will survive. These dynamics also involve survival rates, which we will discuss further, showing how they affect the numbers of calves that mature into yearlings and subsequently into adults. By recognizing these patterns, ranchers can better anticipate future herd sizes, ensuring sustainable management of their resources. Population dynamics in cattle are crucial for effective planning and management in agriculture.

Survival Rates

Survival rates are critical in population modeling because they indicate the proportion of individuals that can be expected to live from one age class to the next. In cattle, different age groups have different survival rates, and understanding these helps predict future population sizes.

• Calves: Only 75% of calves survive to become yearlings. This means that out of every 100 calves, 75 are expected to make it to their first birthday.
• Yearlings: Approximately 80% of yearlings survive to become adults. This slightly increased survival rate accounts for yearlings being more robust and better able to handle environmental pressures.
• Adults: A substantial 90% of adults survive each year. Adults are the hardiest group, having already weathered the vulnerable early years.

These survival rates are vital for estimating how many cattle will advance to the next age class each year, informing management decisions such as how many cattle can be sold or culled without compromising herd sustainability.

Age Class Structure

Age class structure in a cattle population refers to the distribution of animals across different age groups. In our problem, cattle are categorized as calves, yearlings, and adults. This structure reflects both biological processes and management strategies, as it helps in forecasting future population change.

Important considerations for age class structures include:

• Balancing the number of each age class for sustainable herd growth.
• Ensuring there are enough adults to keep the birth rate steady, while maintaining enough young cattle to replace aging adults.
• Managing the population to prevent overstocking and underutilization of resources.

Each age class plays a distinct role in the population dynamics. Calves are future breeders, yearlings are transitional animals crucial for steady population turnover, and adults are the main breeding group. A healthy age class structure supports robust population dynamics and reflects good management practices.

Population Estimation in Ecology

Population estimation in ecology focuses on predicting the number of individuals in a population at any given time, helping in understanding ecological balance and management needs. Various methods are employed, often involving mathematical models like the one applied in our cattle scenario.

Key steps in population estimation involve:

• Calculating birth contributions from adults.
• Incorporating survival rates for each age class.
• Adjusting numbers retrospectively to deduce previous population sizes.

In the provided exercise, we estimated the population for the next and previous spring by using known survival rates and birth contributions. This approach provides insights into how populations evolve over time and helps in devising strategies for future resource management.

Ecological studies often rely on these techniques to not only understand how a specific population is faring but also to insightfully integrate this knowledge into broader ecosystem management and conservation efforts.