Yes and no.
"Natural increase" is births minus deaths.
"Population growth" includes Immigration and emigration too!
So populations can have a negative natural increase (more deaths than births) but still have population growth because immigrants enter the country and settle there permanently (migration). Case in point - the USA.
If birth and death rates are equal there'll be no growth. This then would be zero population growth (ZPG). Don't worry, it's coming, you'll just have to wait a century.
There is a change in population.
When the death rate and the birth rate are equal. This basically means that the amount of people that are dying are being replaced by babies being born. But no extra babies are being born so there is no population growth.
It achieves what used to be called "ZPG" - zero population growth. Births and deaths are essentially equal. The population sustains itself where it is, but does not grow or shrink.
There are different versions of this. One version of zero growth is when the population is at replacement levels. This is actually difficult to calculate exactly because of changing child mortality rates, changes in marriage and death rates, and so on. Another version is when the population dies off -- zero growth!
Population stabilization: Common term for zero population growth, in which the birth rate equals the death rate and in addition where net immigration equals net emigration so that the population does not increase or decrease over time. Usually used in the context of stabilizing increasing populations.http://www.susps.org/overview/population_terms.html
When the death toll and birth rate of people are equal around the world.
It doesn't change.
Birth and immigration both add individuals to a population, increasing the population size. Similarly, deaths and emigration remove individuals, reducing the population. So growth would be equal to the sum of immigration and births, minus the sum of emigration and deaths.
In demographics stationary population is when the death rate and birth rate are equal therefore the population is neither decreasing or increasing but it is stationary.
If immigration and emigration numbers remain equal, the most important contributing factor to a slowed growth rate would likely be a decrease in the birth rate of the population. With births being the primary driver of population growth in this scenario, a decline in the birth rate would result in a slower overall growth rate.
Zero population growthThe so-called replacement rate is about 2.2 births per fertile woman. That's what a society needs to maintain zero population growth. That "extra" 0.2 people is what's needed to account for premature deaths -- deaths that occur before procreation. Zero population occurs when the numbers of birth rate and death rate are the same/equal.