Differential Equations

P(t) = 5000e^{0.08t}

P(t) = 5000 + 0.08t

P(t) = 5000e^{8t}

P(t) = 5000e^{0.08}

The solution $A(t)$ describes exponential growth if $r>0$ or decay if $r<0$.

$A(t)$ will eventually approach a limiting value regardless of $r$.

The solution represents a linear function when $r=0$.

If $r$ is proportional to $t$, the graph of $A(t)$ would be parabolic.

$p(t) = e^{0.3t} + 200$

$p(t) = 200e^{-0.3t}$

$p(t) = 200e^{0.3t}$

$p(t) = \frac{1}{0.3}\ln(200t)$

$y = 2e^{5x} + 2$

$y = 5e^{2x}$

$y = 2e^{5x} - 2$

$y = 2e^{5x}$

Carrying capacity of the environment for the population

Exponential growth rate without environmental limitations

Rate at which new individuals are born

Initial size of the population at time t=0

$\frac{dP}{dt} = 0.03P$

$\frac{dP}{dt} = 0.03t$

$\frac{dP}{dt} = 3t$

$\frac{dP}{dt} = 3P$

Whenever external interventions distort regular progression cycles, undermining the integrity of the foundational elements crucial for supporting the long-term viability initially established in the presets.

When N becomes equal to the baseline comparison term $N_0$, complicating things logarithmically-wise and relationless from an overall perspective of the effectivity designated by the outset norms.

When outward forces exceed the inherent natural repopulation rates, thereby forcing a revaluation of the base figures previously regarded as static environmentals.

During phases of rapid expansion or depletion that surpasses the ability of the systems to compensate accordingly, making it imperative to reassess the grounding benchmark indices.

The population will decrease to zero.

The population growth rate will approach zero.

The population will stabilize at a certain carrying capacity.

The population will grow without bounds.

Neither increase nor decrease has a direct proportional relation to fish population growth rate

For any value of b, increasing it will always result in a rapid increase in growth

As b increases, the growth rate decreases, provided that N does not change simultaneously

b has no effect on the growth rate, since the population number remains constant

y = C(0.02x)

y = e^{0.02x}

y = 0.02x + C

y = Ce^{0.02x}