"I am trying to prove f1(t−T1)∗f2(t−T2)=c(t−T1−T2)f1(t−T1)∗f2(t−T2)=c(t−T1−T2) given that f1(t)∗f2(t)=c(t)f1(t)∗f2(t)=c(t). Here f1(t)∗f2(t)=∫[infinity]−[infinity]f1(x)f2(t−x)dxf1(t)∗f2(t)=∫[infinity]−[infinity]f1(x)f2(t−x)dx.
I can only deduce that f1(t−T1)∗f2(t)=c(t−T1)f1(t−T1)∗f2(t)=c(t−T1) and f1(t)∗f2(t−T2)=c(t−T2)f1(t)∗f2(t−T2)=c(t−T2) by directly evaluating the integral and using the commutative property. According to my textbook, the conclusion should follows directly from these two relationships but I am not able to prove it.
If the 'left' function is delayed by T1T1 seconds, so is c(t)c(t). The same thing happens when the 'right' function is delayed by T2 seconds. So it is quite 'logical' for c(t) to be delayed by the sum of the delayed periods of time in f1 and f2. But this verbal explanation does not satisfy me.
I am also having trouble to express f1(t−T1)∗f2(t−T2) in integral form. I cannot only express it because the definition of the convolution integral seems limited. Thanks in advance."Considering the provided information on the Leslie matrix, dominant eigenvalue, and age structure of a deer population, what can be inferred about the population's dynamics?
a) The population is expected to grow over time since the dominant eigenvalue is greater than one.
b) The population is expected to decline over time as the dominant eigenvalue is less than one.
c) The age structure indicates an unstable population, contrary to the stable age distribution results.
d) The calculated eigenvalues and age structures do not provide sufficient information to predict population dynamics.