(a) Define regression. Explain different methods and applications of regression analysis. (20 marks) (b) Define food chain…

(a) Regression: definition, methods and applications

Definition: regression is the statistical study of how a dependent variable (Y) changes with one or more independent variables (X), expressed as an equation that predicts Y from X. Unlike correlation, which measures only the strength and direction of association, regression gives the form of the relation. Methods: Simple linear regression: Y = a + bX, where the slope b = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)² is the change in Y per unit X and a is the intercept. The constants are found by the method of least squares, which minimises the sum of squares of deviations of observed Y from the line. There are two lines of regression, Y on X and X on Y, and they cut at the means. Multiple regression: Y = b₀ + b₁X₁ + b₂X₂ + …, which separates the effects of several factors. Polynomial (curvilinear) regression: Y = a + bX + cX², for relations such as growth curves that bend. Logistic regression: logit(p) = b₀ + b₁X, used when the outcome is binary, such as presence or absence of a species. Goodness of fit: the coefficient of determination R² gives the proportion of variance in Y explained, and the standard error of the estimate and the significance of b are tested by t or F tests. Applications: Ecology: predicting species distribution from temperature, rainfall and altitude, and the effect of climate change on range. Physiology and growth: relating body weight to length or age, and metabolic rate to body size. Epidemiology: relating disease incidence to climate or vector density. Fisheries and wildlife: predicting population and yield from catch and habitat data. Genetics: heritability is estimated from regression of offspring on parent.

(b) Food chain: definition and types

Definition: a food chain is the linear sequence of organisms through which food energy passes, from producers to herbivores to carnivores, each stage being a trophic level. Only about 10 per cent of energy is transferred to the next level, so chains are short, usually of four or five links. Types: Grazing food chain: begins with living green plants and runs through herbivores to carnivores. Examples: grass, grasshopper, frog, snake, hawk in a grassland; phytoplankton, zooplankton, small fish, large fish in a pond. Detritus food chain: begins with dead organic matter that is eaten by detritivores and decomposers. Examples: leaf litter, earthworm, centipede, shrew on a forest floor; mangrove leaf litter, crab, fish, fishing cat in the Sundarbans. It is the major energy path in forests and estuaries. Parasitic food chain: runs from large host to smaller parasite and hyperparasite, so that the numbers increase along it. Example: a tree, caterpillar, parasitic wasp, hyperparasitic wasp. Interlinking of many chains forms a food web, because most animals eat more than one kind of food.

(c) Altruism and kinship in animal behaviour

Altruism: behaviour that costs the actor (C) in survival or reproduction while it benefits the recipient (B). It seemed to conflict with natural selection, since the donor's fitness falls. Kin selection: Hamilton explained it by inclusive fitness, the sum of an individual's own offspring and the effect on relatives who share genes. Altruism spreads if rB > C, where r is the coefficient of relatedness between donor and recipient. Examples from kin: Alarm calls: a Belding's ground squirrel or meerkat that calls draws the predator to itself, but warns relatives. Cooperative breeding: helper bee-eaters delay their own breeding to feed the young of parents or siblings. Social insects: sterile worker honeybees work for the queen. Sisters share 0.75 of their genes by haplodiploidy when the queen mates once, but where she mates with many males the relatedness is lower, so colony-level benefits also matter. Reciprocal altruism: between non-relatives, helping is repaid later, as when vampire bats share blood meals with roost-mates that fed them before. It needs repeated meetings and recognition of cheats.