The Population Logit Model of the Binary Dependent Variable Y $$\operatorname { Pr } \left( Y = 1 \mid X _ { 1 } \right) = \frac { 1 } { 1 + e ^ { - \left( \beta _ { 0 } + \beta _ { 1 } X _ { 1 } \right) } }$$

question 1

Essay

The population logit model of the binary dependent variable Y with a single regressor is $$\operatorname { Pr } \left( Y = 1 \mid X _ { 1 } \right) = \frac { 1 } { 1 + e ^ { - \left( \beta _ { 0 } + \beta _ { 1 } X _ { 1 } \right) } }$$ Logistic functions also play a role in econometrics when the dependent variable is not a
binary variable.For example, the demand for televisions sets per household may be a
function of income, but there is a saturation or satiation level per household, so that a
linear specification may not be appropriate.Given the regression model $$Y _ { i } = \frac { \beta _ { 0 } } { 1 + \beta _ { 1 } e ^ { - \beta _ { 2 } X _ { i } } } + u _ { i } ,$$ sketch the regression line.How would you go about estimating the coefficients?

Distinguish between direct and indirect competition in ecological communities.

Explain the concept of niches and the effect of species interactions on niche dimensions.

Identify the role and characteristics of keystone species within ecological communities.

Recognize the importance of mutualism and its impact on species interactions and community structure.